Wavelet Based Approximation Schemes For Singular Integral Equations 1st Edition Madan Mohan Panja

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Wavelet Based
Approximation Schemes for
Singular Integral Equations
M M Panja
Department of Mathematics
Visva-Bharati, Santiniketan, India
B N Mandal
Professor (retired), Physics and Applied Mathematics Unit
Indian Statistical Institute, Kolkata, India
A SCIENCE PUBLISHERS BOOK
p,
A SCIENCE PUBLISHERS BOOK
p,

CRC Press
Taylor & Francis Group
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Preface
The mathematical modelling of physical processes and the computational mathematics have complemented
each other since the era of Newton. The underlying principle of numerical technique in computational
mathematics was the interpolation based on the solid foundation, the Weierstrass approximation theorem.
It is well known that here unknown functions are represented/approximated in polynomial basis whose
coef&#6684777;cients are determined by the values of the unknown function prescribed at some points within its domain
of de&#6684777;nition. But around 1900, mathematician Runge observed that approximation based on interpolation
scheme is unable to represent functions ef&#6684777;ciently which are continuous and even differentiable within its
domain of de&#6684777;nition.
On the other hand after the development of the formal theory of function spaces, another computational
scheme known as Fourier approximation has been developed. Here the unknown functions are approximated/
represented by the linear combination of harmonics with coef&#6684777;cients involving integrals of the unknown
function and the corresponding harmonics. This scheme is found to be well suited for approximating unknown
solutions of differential and integral equations (arising in the mathematical analysis of physical processes)
which are smooth enough within the domain of interest.
Again, around 1900, J W Gibbs pointed out that approximation based on harmonics (trigonometric
function, in particular) is unable to represent functions having &#6684777;nite discontinuities in their domain. Over
and above, estimation of error in approximation of function in the numerical methods based on classical
harmonics requires exhaustive mathematical analysis.
It is thus desirable to search for a computational scheme which can effectively approximate functions
which are smooth in most of the region but may have sharp variations within a narrow region, even may
have &#6684777;nite/in&#6684777;nite discontinuities within the domain of interest as well as provide a posteriori error in a
straightforward way.
One of our objectives here is to present a computational scheme based on a novel mathematical structure,
known as multiresolution analysis (MRA) of function space which may be regarded as the con&#6684780;uence of
several existing computational schemes as well as a mathematical microscope. We will concentrate here on
the L
2
-space or its subspace. The scheme of our presentation is as follows:
• An overview of MRA of L
2
(R)/L
2
([a, b])
• Multiresolution approximation of functions and operators in L
2
(R)/L
2
([a, b])
• Wavelet based computational schemes for getting approximate solution of integral equations of second
kind with singular kernels, in particular.
In many &#6684777;elds of application of mathematics, progress is crucially dependent on the good &#6684780;ow of
information between (i) theoretical mathematicians looking for applications, (ii) mathematicians working on
applications in need of theory, and (iii) scientists and engineers applying mathematical models and methods.
The intention of this book is to stimulate this &#6684780;ow of information.
In the &#6684777;rst chapter some mathematical prerequisites of singular integral equations have been presented.
The underlying mathematical structure of wavelet bases as desired in this monograph have been described
in chapter two. In chapter three, mathematical formulae and tricks for approximation of functions,
representation of (differential and integral) operators have been described in somewhat details. The ef&#6684777;ciency

iv <Wavelet Based Approximation Schemes for Singular Integral Equations

of the formulae derived here have been tested through their applications to the relevant test problems. The
knowledge and techniques developed in the last two chapters have been applied in subsequent chapters for
obtaining approximate solutions of Fredholm integral equation of second kind with a variety of singular
kernels. In chapter four we have considered weakly singular kernel with both logarithmic and algebraic types.
Chapter 5 deals with a Fredholm integral equation of second kind with a special type of kernel having
singularity at a &#6684777;xed point. Fredholm integral equation of second kind with Cauchy singular kernels in
both bounded and unbounded domain have been studied in Chapter 6. The singular integral equation with
hypersingular kernels have been discussed in Chapter 7. Some numerical data for several ingredients
involved in the wavelet based numerical scheme for obtaining approximate solution of integral equations,
with singular coef&#6684777;cients or kernels, in particular have been presented in the Appendices.
The authors thank Dr. Swaraj Paul for providing some material of this book prepared jointly during
the tenure of his Ph.D. MMP is thankful to Dr. Prakash Das, Debabrata, Sayan, and Mouzakkir for their
participation in preparation of some results and help in typing of this monograph. He is deeply indebted to
his family members, wife Manju, brother Amit, children Dibya, Rohini and Rivu in particular, who provided
him with their continued encouragement, patience and support during the preparation of this book.
The authors would highly appreciate any correspondence concerning constructive suggestion.
Finally, the authors thank Mr. Vijay Primlani of Science Publishers (CRC Press) for his support and
patience in the preparation of the monograph.
M M Panja
B N Mandal

Preface iii
1. Introduction 1
1.1 Singular Integral Equation 1
1.1.1 Approximate solution of integral equations 2
1.1.1.1 The general scheme of approximation 3
1.1.1.2 Nyström method 4
1.1.1.3 Collocation method 6
1.1.1.4 Galerkin’s method 7
1.1.1.5 Quadratic spline collocation method 9
1.1.1.6 Method based on product integration 10
1.1.2 Kernel with weak (logarithmic and algebraic) singularity 11
1.1.3 Integral equations with Cauchy singular kernel 13
1.1.3.1 Method based on Legendre polynomials 13
1.1.3.2 Method based on Chebyshev polynomials 14
1.1.3.3 Method based on Jacobi polynomials 15
1.1.4 Integral equations with hypersingular kernel 20
2. Multiresolution Analysis of Function Spaces 22
2.1 Multiresolution Analysis of L
2
(R) 23
2.1.1 Multiresolution generator 23
2.1.2 Wavelets 23
2.1.3 Basis with compact support 24
2.1.4 Properties of elements in Daubechies family 25
2.1.5 Limitation of scale functions and wavelets in Daubechies family 27
2.2 Multiresolution Analysis of L
2
([a, b] ⊂ R) 27
2.2.1 Truncated scale functions and wavelets 28
2.2.2 Multiwavelets 33
2.2.3 Orthonormal (boundary) scale functions and wavelets 36
2.3 Others 40
2.3.1 Sinc function 40
2.3.2 Coi&#6684780;et 43
2.3.3 Autocorrelation function 47
Contents

vi <Wavelet Based Approximation Schemes for Singular Integral Equations

3. Approximations in Multiscale Basis 50
3.1 Multiscale Approximation of Functions 50
3.1.1 Approximation of f in the basis of Daubechies family 51
3.1.1.1 f ∈ L
2
(R) 51
3.1.1.2 Orthonormal basis for L
2
([a, b]) 53
3.1.1.3 Truncated basis 55
3.1.2 Approximation of f ∈ L
2
([0, 1]) in multiwavelet basis 56
3.2 Sparse Approximation of Functions in Higher Dimensions 58
3.2.1 Basis for Ω ⊆ R
2
58
3.2.1.1 Representation of f (x, y) 60
3.2.1.2 Homogeneous function K (λx, λy) = λ
μ
K (x – y), μ ∈ R 61
3.2.1.3 Non-smooth function f (x, y) = |x + y|
ν
, ν ∈ R – {N ⋃ 0} 63
3.2.1.4 f (x, y) = ln|x ± y| involving logarithmic singularity 63
3.2.1.5 f ∈ Ω ⊂ R
2
64
3.3 Moments 69
3.3.1 Scale functions and wavelets in R 69
3.3.2 Truncated scale functions and wavelets 69
3.3.3 Boundary scale functions and wavelets 70
3.4 Quadrature Rules 71
3.4.1 Daubechies family 71
3.4.1.1 Nodes, weights and quadrature rules 73
3.4.1.2 Formal orthogonal polynomials, nodes, weights of scale functions 74
3.4.1.3 Interior scale functions 74
, Φ
right
on R
–
)3.4.1.4 Boundary scale functions (Φ
left
on R
+
78
3.4.1.5 Truncated scale functions (Φ
LT
, Φ
RT
on [0, 2K – 1]) 80
3.4.1.6 Formal orthogonal polynomials, nodes, weights of wavelets 80
3.4.1.7 Algorithm 83
3.4.1.8 Error estimates 85
3.4.1.9 Numerical illustrations 86
3.4.2 Quadrature rules for singular integrals 91
3.4.2.1 Integrals with logarithmic singularity 92
3.4.2.2 Quadrature rule for weakly (algebraic) singular integrals 96
3.4.2.3 Quadrature rule for Cauchy principal value integrals 100
3.4.2.4 Finite part integrals 105
3.4.2.5 Composite quadrature formula for integrals having Cauchy and 106
weak singularity
3.4.2.6 Numerical examples 108
3.4.3 Logarithmic singular integrals 110
3.4.4 Cauchy principal value integrals 111
3.4.5 Hypersingular integrals 111
3.4.6 For multiwavelet family 113

3.4.7 Others 114
3.4.7.1 Sinc functions 114
3.4.7.2 Autocorrelation functions 115
3.4.7.3 Representation of function and operator in the basis generated by 116
autocorrelation function
3.5 Multiscale Representation of Differential Operators 117
3.6 Representation of the Derivative of a Function in LMW Basis 118
3.7 Multiscale Representation of Integral Operators 121
3.7.1 Integral transform of scale function and wavelets 121
3.7.2 Regularization of singular operators in LMW basis 125
3.7.2.1 Principle of regularization 125
3.7.2.2 Regularization of convolution operator in LMW basis 125
3.8 Estimates of Local Hölder Indices 126
3.8.1 Basis in Daubechies family 126
3.8.2 Basis in Multiwavelet family 126
3.9 Error Estimates in the Multiscale Approximation 128
3.10 Nonlinear/Best n-term Approximation 134
4. Multiscale Solution of Integral Equations with Weakly Singular Kernels 135
4.1 Existence and Uniqueness 135
4.2 Logarithmic Singular Kernel 137
4.2.1 Projection in multiscale basis 137
4.2.1.1 Basis in Daubechies family 137
4.2.1.2 LMW basis 138
4.3 Kernels with Algebraic Singularity 143
4.3.1 Existence and uniqueness 143
4.3.2 Approximation in multiwavelet basis 143
4.3.2.1 Scale functions 144
4.3.2.2 Scale functions and wavelets 147
4.3.2.3 Wavelets 148
4.3.2.4 Multiscale approximation (regularization) of integral operator K
A
in 150
LMW basis
4.3.2.5 Reduction to algebraic equations 150
4.3.2.6 Multiscale approximation of solution 152
4.3.2.7 Error Estimates 152
4.3.3 Approximation in other basis 157
5. An Integral Equation with Fixed Singularity 162
5.1 Method Based on Scale Functions in Daubechies Family 163
5.1.1 Basic properties of Daubechies scale function and wavelets 163
5.1.2 Method of solution 165
5.1.3 Numerical results 169
Contents <vii

viii <Wavelet Based Approximation Schemes for Singular Integral Equations

6. Multiscale Solution of Cauchy Singular Integral Equations 171
6.1 Prerequisites 172
6.2 Basis Comprising Truncated Scale Functions in Daubechies Family 174
6.2.1 Evaluation of matrix elements 175
6.2.1.1 k , k' ∈ ⋀
V
j
I
176
6.2.1.2
IT LT
k ∈ ⋀
V
j
, k' ∈ ⋀
V
j
176
6.2.1.3
LT LT
k ∈ ⋀
V
j
, k' ∈ ⋀
V
j
177
6.2.1.4
LT RT
k ∈ ⋀
V
j
, k' ∈ ⋀
V
j
178
6.2.1.5
IT RT
k ∈ ⋀
V
j
, k' ∈ ⋀
V
j
178
6.2.1.6
RT RT
k ∈ ⋀
V
j
, k' ∈ ⋀
V
j
179
6.2.2 Evaluation of f
T
179
j
6.2.3 Estimate of error 180
6.2.4 Illustrative examples 180
6.3 Multiwavelet Family 184
6.3.1 Equation with constant coef&#6684777;cients 184
6.3.1.1 Evaluation of integrals 185
6.3.1.2 Multiscale representation (regularization) of the operator K
C
in 188
LMW basis
6.3.1.3 Multiscale approximation of solution 188
6.3.1.4 Estimation of error 189
6.3.1.5 Illustrative examples 190
6.3.2 Cauchy singular integral equation with variable coef&#6684777;cients 193
6.3.2.1 Evaluation of integrals involving function, Cauchy singular kernel and 193
elements in LMW basis
6.3.2.2 Evaluation of the integrals involving product of a(x), scale functions 198
and wavelets
6.3.2.3 Multiscale representation (regularization) of the operator ωK
C
in 199
LMW basis
6.3.2.4 Multiscale approximation of solution 200
6.3.2.5 Estimate of Hölder exponent of u(x) at the boundaries 201
6.3.2.6 Estimation of error 201
6.3.2.7 Applications to problems in elasticity 202
6.3.3 Equation of &#6684777;rst kind 208
6.3.3.1 Evaluation of integrals involving kernel with &#6684777;xed singularity and 210
elements in the LMW basis
6.3.3.2 Evaluation of integrals involving kernel with &#6684777;xed singularity and 215
weight factor
6.3.3.3 Multiscale representation (regularization) of the operator ωK
F
in 220
LMW basis
6.3.3.4 Multiscale approximation of solution 221
6.3.3.5 Illustrative examples 222
6.3.4 Autocorrelation function family 230

6.3.5 In R 233
6.3.5.1 Transformation to the &#6684777;nite range of integration 234
6.3.5.2 Multiscale approximation of solution 236
6.3.5.3 Estimation of error 236
6.3.5.4 Illustrative examples 237
6.3.6 Other families 239
6.3.6.1 Hilbert transform 239
6.3.6.2 Integral equation of second kind 241
7. Multiscale Solution of Hypersingular Integral Equations of Second Kind 244
7.1 Finite Part Integrals Involving Hypersingular Functions 244
7.2 Existing Methods 246
7.3 Reduction to Cauchy Singular Integro-differential Equation 247
7.4 Method Based on LMW Basis 248
7.4.1 Multiscale approximation of the solution 249
7.4.2 Estimation of error 250
7.4.3 Illustrative examples 250
7.5 Other Families 254
Appendices 258
References 269
Author Index 283
Subject Index 285
Contents <ix

Chapter 1
Introduction
1.1 Singular Integral Equation
In mathematics, singular integrals and integral operators with singular kernels have a well-established
theoretical basis (Muskhelishvili, 2013; Mikhlin, 2014). For example, the weakly singular (WS)
integrals are considered as improper integrals, the singular integrals are considered in the sense of
Cauchy principal values (CPV) and the hypersingular integrals integrals are considered in the sense
of Hadamard ﬁnite parts (FP) (Muskhelishvili, 2013; Mikhlin, 2014; Kanwal, 1998). As for example,
for a < x < b,

b

WS ln| − y|

x dy = lim

x
Mandal
NBandPanjaM
−E
−
b
Mln|x y|dy + ln x y dy
→
a x+ E
E0
a
| − |
= (b − x)ln(b − x) + (x a)ln(x

a) (b a)

1
WS

− − − − (1.1.0.1a)
b

dy = lim
−
x

x E1 1 b 1
| − y|
µ
→ →
a |x −y |
µ dy +

µ dy
E
x +E
0,E0
2 x y
a
1 2
| − |
(x −

a)
1−µ
(b −x)
1−

µ
= +

1−µ 1−µ
0< µ <1 (1.1.0.1b)
b
1
CPV

x −E 1

b
dy = lim dy +
1
ax

dy
a x−y x +Ex −y
− y E →0
= (ln(b

− x) − ln(x − a)) (1.1.0.1c)
b
1 x E b
FP

dy = lim

− 1
dy +
1

2
(x− y)
2 a (x −y )
2

x +E (x −y
a
→
2dy
)
−
E

E 0
1 1
=

− +

(1.1.0.1d)
b−x a−x
The theory of distributions (generalized functions) lets us to consider divergent integrals and
integral operators with kernels containing diﬀerent kinds of singularities in the same approach of
omission of singular parts (Kanwal, 1998) as followed in the formula (1.1.0.1d). The divergent
integrals must be evaluated when the boundary integral equation (BIEs) are solved numerically
using the boundary element methods (BEM). There are several methods for the evaluation of the
weakly singular and singular integrals (Muskhelishvili, 2013; Mikhlin, 2014; Mandal and Chakarbarti,
2016). Hypersingular integrals are more complex and there are some problems with their numerical
calculation. Therefore, the BIE with singular integrals (in the sense of CPV) have been used until

M
M Panja and B N Mandal

2 1.1. Singular Integral Equation
recently. However, there are some kinds of problems where the BIE with hypersingular integrals are
preferable and closer to the physical sense of the problem. Such a situation takes place in the theory
of elasticity and fracture mechanics when the BIE method is used to solve problems for bodies with
cuts and cracks.
It was observed by Gantumur and Stevenson (Gantumur and Stevenson, 2006), that boundary
integral methods reduce elliptic boundary value problems to integral equations formulated on the
boundary of the domain. Although the dimension of the underlying manifold decreases by one,
the ﬁnite element discretization of the resulting boundary integral equation gives densely populated
stiﬀness matrices, causing serious obstructions to accurate numerical solution processes. In order
to overcome this diﬃculty, various successful approaches for approximating the stiﬀness matrix
by sparse ones have been developed, such as multipole expansions, panel clustering, and wavelet
compression (see, e.g. (Atkinson, 1997; Hackbusch, 1995)).
In their work, Beylkin, Coifman and Rokhlin (Beylkin et al., 1991) ﬁrst observed that wavelet
bases give rise to almost sparse stiﬀness matrices for the Galerkin discretization of singular integral
operators, meaning that the stiﬀness matrix has many small entries that can be discarded without
reducing the order of convergence of the resulting solution. This result initiated the development
of eﬃcient compression techniques for boundary integral equations based upon wavelets. In their
studies Dahmen, Harbrecht and Schneider (Dahmen et al., 2006; Dahmen et al., 2007) showed
that for a wide class of boundary integral operators a wavelet basis can be chosen so that the
full accuracy of the Galerkin discretization can be retained at a computational work of order N
(possibly with a logarithmic factor in some studies), where N is the number of degrees of freedom
used in the discretization. First nontrivial implementations of these algorithms and performance
tests were reported by Lage and Schwab (Lage and Schwab, 1999; Lage and Schwab, 2001). The
main reason why a stiﬀness matrix entry is small is that the kernel of the involved integral operator is
increasingly smooth away from its diagonal, and that the wavelets have vanishing moments, meaning
that they are L
2
-orthogonal to all polynomials up to a certain degree. Another advantage of a
wavelet-Galerkin discretization is that the diagonally scaled stiﬀness matrices are well-conditioned
uniformly in their sizes, guaranteeing a uniform convergence rate of iterative methods for the linear
systems. Finally, recent developments suggest a natural use of wavelets in adaptive discretization
methods that approximate the solution using, up to a constant factor, as few degrees of freedom as
possible.
1.1.1 Approximate solution of integral equations
To solve approximately the integral equation

u(x) + λ K(x, s)u(s)ds = f(x), x ∈ Ω � R, (1.1.1.1)
Ω
choose a ﬁnite dimensional family of functions that is believed to contain a function uapprox(x) close
to the true solution u(x). The desired numerical (approximate) solution uapprox(x) is selected by
having it to satisfy (1.1.1.1) approximately. There are various senses in which uapprox(x) can be said
to satisfy (1.1.1.1) approximately, and these lead to provide space for diﬀerent types of approximation
methods. Such methods are mostly of two types, collocation methods and Galerkin methods. Most
of the methods now available are variants of these two. When these methods are formulated in an

M M Panja and B N Mandal

3 1.1. Singular Integral Equation
abstract framework using functional analysis, they all make essential use of projection operators.
Since the error analysis is most easily carried out within such a functional analysis framework, we
call collectively all such methods as suggested by Atkinson, as projection methods (Atkinson, 1997).
The underlying principle of these approximations is the following.
1.1.1.1 The general scheme of approximation
We consider here two Banach spaces X and Y and the (functional) equation
O[u] = f (1.1.1.2)
relating element u ∈ D(O) ⊂ X to an element f ∈ R(O) ⊂ Y. Here O : X → Y is the linear
operator from the Banach space X into Y, D(O) being the domain, R(O) represents the range of
the operator O. The Eq. (1.1.1.2) and the element u ∈ X are termed as the exact equation and the
exact solution respectively. We use the symbol L(X, Y) as the space of linear operators mapping
from X into Y.
We further introduce sequence of projection operators P
h : X → X h, P
�
: Y → Y h such that
h
Xh ⊂ X, P hX = X h, P
2
= Ph (1.1.1.3a)
h
Yh ⊂ Y, P
h
� Y = Y h, P
�2
= P
�
(1.1.1.3b)
h h
where X h ⊂ X and Y h ⊂ Y are ﬁnite dimensional subspaces of the Banach spaces X and Y
respectively. Here the symbol h ∈ R (a dyadic proper fraction here) represents the parameter for
discretization.
Now consider A
h : Xh → Y h, a mapping in L(X h, Yh) and an approximate equation
A
h[uh] = f h (1.1.1.4)
with
A
h = P
h
� O Ph, uh = Phu, f h = P
h
� f (1.1.1.5)
obtained by using projection operators P
h and P
�
mentioned above.
h
Deﬁnition 1.1. The solution u h of the approximate Eq. (1.1.1.4) is called approximate solution of
Eq. (1.1.1.2).
This scheme can be represented through the following diagram.
O
X ⊃ ⊂ Y
P
h
⏐
⏐
⏐
⏐
�
⏐
⏐
⏐
⏐
�
D(O) −→ R(O)
P
�
h
(1.1.1.6)
Ah
Xh ⊃ D(A h) −→ R(A h) ⊂ Y h
The existence of exact solution to (1.1.1.2), convergence of the approximate solution of (1.1.1.4) to
the exact one and the stability of approximations are the main points to have been addressed in the
application of this scheme. To take into account these issues, it is convenient to put these steps in
some mathematical set up.

M M Panja and B N Mandal

4 1.1. Singular Integral Equation
It is assumed that projection operators P
h and P
h
U

converge to identity operators in X and Y
respectively, i.e.,
lim
hu u= 0 u X (1.1.1.7a)
h→0
P − ∀ ∈
lim P
h
Uf − f = 0 ∀ f ∈ Y. (1.1.1.7b)
h→0
Theorem 1.2. If
i) the projection operators P
h and P
h
U

satisfy the conditions in (1.1.1.7),
ii) the sequence of approximate operators A
h converges to O on each exact solution, i.e.,
lim A
h[uh] − O[u] Y= 0, (1.1.1.8)
h→0
iii) the condition
A
h[uh] − O[u] Y≥ γ u h Xh
∀uh ∈ Xh (1.1.1.9)
of stability is satisﬁed for the sequence of operators {A
h},
then
a) the exact solution of (1.1.1.2) exists and unique,
b) a unique solution u
h of the approximate Eq. (1.1.1.4) exists for all h small enough,
c) the sequence {u
h} of approximate solutions converges to the exact one and takes place in the
estimation
u
h − Phu Xh
≤ P
h
U O[u] − A h[Phu] Yh
. (1.1.1.10)
Thus, in the approximation methods based on projection, instead of searching exact solution
of (1.1.1.2) in the space X of functions, one is intended to ﬁnd a sequence of solutions u
h of the
approximate Eq. (1.1.1.4) in ﬁnite dimensional (projection) space X
h and continue to decrease h
until the desired accuracy has been achieved. The spaces of functions {X, X
h} and {Y, Y h} are
related by means of the projection operators P
h ∈ L(X, X h) and P
U
∈ L(Y, Y h) respectively.
h
1.1.1.2 Nystr¨om method
The Nystr¨om method was introduced to handle approximations based on numerical integration of
the integral equation

a u(x) − b K(x, s) u(s) ds = f(x), x ∈ D. (1.1.1.11)
D
The solution is found ﬁrst at the set of quadrature node points used for the evaluation of the integral
present in the equation, and then such values extended to other points in D by means of interpolation
formula. The numerical method is much simpler to implement on a computer while the error analysis
is more sophisticated. In this section, basic principle of Nystr¨om method, which assumes the kernel
function to be integrable have been presented. A preliminary error analysis has also been presented
here.

5 1.1. Singular Integral Equation
Let a numerical integration scheme
n

f(x) dx
D
≈ ωi f(xi) (1.1.1.12)
i=1
be given where xi(i = 1, · · ·, n) and ωi(i = 1,
n
· · ·, n) are nodes and weights respectively.
It is assumed that the kernel K(x, s) is integrable ∀x, s ∈ D, where the domain Dis a closed
and bounded set. Using the quadrature scheme (1.1.1.12) mentioned above one can approximate
the integral in (1.1.1.11) and obtain a new equation
n
n

aun(x) −b ω iK(x, si) un(si) = f(x), x ∈ D. (1.1.1.13)
i=1
We regard this as an exact equation with a new unknown function un(x). To ﬁnd the solution at the
nodes (si, i = 1, · · ·, n), let x run through the quadrature node points xk, k = 1, · · ·, n. This yields
a system of linear equations
n

a un(xk) −b
n
K(xk, si) un(si) = f(xk), k = 1, , n (1.1.1.14)
i=1
· · ·
for the unknown un = (un(x1), un(x2), · · ·, un(xn)). Assuming [aδk,i −bK(xk, si)] w
n×n
is a ell
conditioned matrix, to each solution u¯n of (1.1.1.14), there is a unique solution of (1.1.1.13) that
agrees with un at the node points. Then, if one solves (1.1.1.13) for un(x) for any x ∈ D, un(x)
may be determined at x (beyond nodes) by its values at the node points xi, i = 1, · · ·, n by using
the formula
1
n
un(x) =

f(x) + b
n
Mandal
NBandωiK(x, si) un(s
Panja
i)
M
M

x
i=1
∈ D. (1.1.1.15)
a
This relation may be regarded as an interpolation formula since
n
1
un(xi) − [f(xi) + b ωiK(xi, si) un(si) = 0 (1.1.1.16)
a
i=1
identically. Formula (1.1.1.15) is known as Nystr
n
om¨ interpolation formula.
Deﬁnition 1.3. For u ∈C[D], corresponding to the integral operator K[u](x) =

K(x, s) u(s) ds,
D
the rule of correspondence
n

Kn[u](x) ≡ ωiK(x, si) un(si), x ∈ D (1.1.1.17)
i=1
is known as numerical integral operator.
n
Observation 1. The operator Kn : C[D] →C[D] is a bounded, ﬁnite rank, linear operator with
n

Kn = max |ωiK(x, si)|. (1.1.1.18)
x∈D
i=1
Observation 2.(Atkinson and Han, 2009, p.104)
n

lim K − Kn
n→∞
→0,
and
K ≤ K − K n .

� �
M M Panja and B N Mandal

6 1.1. Singular Integral Equation
1.1.1.3 Collocation method
It is ﬁrst assumed that the approximate solutions u
n(x), n ∈ N of the exact solution u(x) are
elements of some ﬁnite dimensional space of functions, may be regarded as the linear span of a basis
{φ
1(x), φ2(x), · · · , φ n(x)}. The underlying principle is to pick distinct node points x 1, x2, ...., x n ∈
D so that the residues
r
n(xi) ≡ λ u n(xi) − K(x i, t)un(t) dt − f(x i) = 0, i = 1, 2, · · · , n. (1.1.1.19)
D
This leads to determining n unknown coeﬃcients {c 1, c2, ...., c n} as the solution of the linear system
d
n
c
j λφj (xi) − K(x i, s)φj (s) ds = f(x i) i = 1, 2, · · · , n. (1.1.1.20)
D
j=1
An immediate question is whether this system has a solution, and if so, whether it is unique. If so,
does u
n converge to u? Note also that, the linear system contains integrals that must usually be
evaluated numerically.
As a part of writing (1.1.1.20) in a more abstract form, we bring here the projection operator
P
n that maps X = C(D) onto X n. Given u ∈ C(D), deﬁne P nx to be that element of X n that
interpolates u at the nodes {x
1, x2, · · · , xn}. This means writing
n
n
P
n u(x) = c kφk(x) (1.1.1.21)
k=1
with the coeﬃcients {c k, k = 1, · · · , n} determined by solving the linear system
n
n
c
kφk(xi) = u(x i), i = 1, · · · , n.
k=1
This linear system has a unique solution if
det[φ
k(xi)] (1.1.1.22) = 0.
Henceforth it is assumed that this is true whenever the collocation method is being used. By a simple
argument, this condition also implies that the elements {φ
1, φ2, ....., φ n} in X n are independent. In
the case of polynomial interpolation (for functions of single variable), the determinant in (1.1.1.22)
is the Vandermonde determinant.
To see more clearly that P
n is linear, and to give a more explicit formula, we introduce a new
set of basis functions. For each k
'
, 1 ≤ k
'
≤ n, let £ k
! ∈ Xn be that element which satisﬁes the
interpolation conditions
£
k
! (xi) = δ k
!
i, i = 1, 2, · · · , n.
By (1.1.1.22), there is a unique such £
i, and the set {£ 1, £2, ...., £ n} is a new basis for X n. With
polynomial interpolation, such functions £
k
! are called Lagrange basis functions, and we will use this
name with all types of approximating subspaces X
n. With this new basis, we can write
n
n
P
nu(x) = u(x i)£i(x), x ∈ D. (1.1.1.23)
i=1

M M Panja and B N Mandal

7 1.1. Singular Integral Equation
Clearly, P
n is linear and is of ﬁnite rank. In addition, as an operator on C(D) to C(D),
n
n
P
n = max | k
! (x)|. (1.1.1.24)
x∈D
k
!
=1
The condition (1.1.1.19) can now be rewritten as
P
nrn = 0 (1.1.1.25)
or equivalently,
P
n(λ − K)u n = Pnf, u n ∈ Xn. (1.1.1.26)
1.1.1.4 Galerkin’s method
Here it is assumed that the solution u is an element of some inner product space, e.g., X = L
2
(D)
or some other Hilbert space and we use the symbol < ·, · > to describe the inner product for X. The
main requirement of the Galerkin method is that the residue r
n satisﬁes
< r
n, φk >= 0, k = 1, 2, · · · , n (1.1.1.27)
instead of condition (1.1.1.25) for the collocation method. The left side is the Fourier coeﬃcient of
r
n associated with the element φ k in the basis of X n. If {φ 1, φ2, ....., φ n} are the leading members of
an orthonormal family Φ ≡ {φ
1, φ2, ....., φ n, ....} that is complete in X, then (1.1.1.27) requires the
leading terms to be zero in the Fourier expansion of r
n with respect to Φ.
nn
To ﬁnd the approximate solution u
n(x) = c k
! φk
! (x), we apply (1.1.1.27) to (1.1.1.19) with
k
!
=1
xi = x. This yields the linear system
n
n
c
k
! {λ < φk
! , φk > − < Kφ k
! , φk >} =< f, φ k >, k = 1, 2, ...., n. (1.1.1.28)
k
!
=1
This is Galerkin’s method for obtaining an approximate solution to (1.1.1.1). Now questions are:
does the system of equations in (1.1.1.28) have any solution? If yes, is it unique? Does the resulting
sequence of approximate solutions u
n converges to u in X ? Does the sequence converge in C(D)?
Note also that, the above formulation contains double integrals < Kφ
k
! , φi >. These must often be
computed numerically. We return to a consideration of this later.
To get answers to questions mentioned above it is convenient to write (1.1.1.28) in an abstract
framework. We introduce here the projection operator P
n that maps X onto X n. For general u ∈ X,
deﬁne P
nu to be the solution of the following minimization problem
u − P
nu = min u − v . (1.1.1.29)
v∈X n
By assumption, X n is ﬁnite dimensional. So, it can be shown that this problem has a solution; and
by employing the fact that X
n is an inner product space, the solution can be shown to be unique.
To obtain a better understanding of P
n, we give an explicit formula for P nu. Introduce a new
basis comprising elements {θ
1, θ2, · · · , θn} for X n by using Gram-Schmidt or any other method to

�
�
M M Panja and B N Mandal

8 1.1. Singular Integral Equation
create an orthonormal basis from {φ
k, k = 1, · · · , n}. The elements {θ k
� , k
�
= 1, 2, · · · , n} are linear
combinations of {φ
1, φ2, ....., φ n}, and moreover
< θ
k, θk
� >= δ k k
� , k, k
�
= 1, 2, · · · , n. (1.1.1.30)
With this new basis, it is straightforward to show that
n
Pnu = < u, θ k > θk. (1.1.1.31)
k=1
This shows immediately that P n is a linear operator. With this formula, we can show the following
results.
2 2 2
�u�= �P nu�+ �u − P nu�, (1.1.1.32)
n
�Pnu�
2
= |(u, θ k)|
2
,
k=1
(Pnu, v) = (u, P nv), u, v ∈ X, (1.1.1.33)
((Id − P
n)u, P nv) = 0, u, v ∈ X. (1.1.1.34)
Because of the last identity, the operator P
nu may be regarded as the orthogonal projection of u ∈ X
onto X
n. Consequently, the operator P n deﬁned by (1.1.1.29) may be regarded as an orthogonal
projection operator. The result (1.1.1.32) leads to
�P
n� = 1. (1.1.1.35)
Using (1.1.1.34), it can be established that
2 2 2
�u − v�= �u − P nu�+ �P nu − v�, v ∈ X n. (1.1.1.36)
This implies that P
nu is the unique solution to (1.1.1.29).
Using the fact that the elements {φ
k, k = 1, · · · , n} of the basis of X n are independent,
P
nz = 0 if and only if (z, φ i) = 0, i = 1, 2, · · · , n. (1.1.1.37)
One can thus rewrite (1.1.1.27) as
P
nrn = 0
or equivalently,
P
n(λ − K)u n = Pnf, u n ∈ Xn. (1.1.1.38)
Note that this relation is similar to (1.1.1.26) appearing in the collocation method.
There is a variant on Galerkin’s method, known as the Petrov-Galerkin method. Here, one
chooses u
n ∈ Xn, but we require
(r
n, w) = 0, ∀ w ∈ W n
where W n is another ﬁnite dimensional subspace of dimension n. This method is not considered
further in this monograph. It is an important method when looking at the numerical solution of
boundary integral equations. Another variant to Galerkin’s method is to set it within a variational
framework.

9 1.1. Singular Integral Equation
1.1.1.5 Quadratic spline collocation method
For n ∈ N, let us consider a grid
Δn = {x0, x1, · · · , xn : 0 = x0 < x1 · · · < xn = 1} (1.1.1.39)
partition of the
(n)
on [0, 1] (a closed interval [0,1] with grid points xi ≡ x
i
, i = 0, 1, · · · , n).
Deﬁnition 1.4. The grid Δn is said to be quasi-uniform if
max (xi+1 xi)
0≤i≤n−1
−

min (xi+1
≤ q (1.1.1.40)
xi)
0≤i≤n−1
−
for some q ≥ 1 independent of n.
Deﬁnition 1.5. The partition is said to be a graded grid if

=

r
x
1 2i
i
2n

i = 0, 1, · · · ,
n

2
,
n (1.1.1.41)
x n
+i = 1 − x n

−i i = 1, 2, · · · , ,
2 2 2
where n ∈ 2N and r ≥ 1 a real number independent of the size of the number of nodes n + 1.
Observation 1. The exponent r present in the deﬁnition characterizes the non-uniformity of the
grid, e.g., the grid is uniform for the choice r = 1 which is densely clustered near the end points 0
and 1.
Observation 2. The graded grid is not quasi-un
Mandal
NBandPanjaM
Miform for r > 1.
Deﬁnition 1.6. The symbol S2,1(Δn) has been used as the collection
S2,1(Δn) = {y(x) ∈ C
1
([0, 1]) : y(x)|
[xi,xi+1]∈ P2, i = 0, 1, · · · , n − 1} (1.1.1.42)
of quadratic splines with defect 1 on the grid Δn mentioned above. Here P2 is the collection of
polynomials of degree not exceeding 2, C
1
([0, 1]) is the set of all continuously diﬀerentiable functions
y in [0, 1].
The explicit variable dependence of elements in S2,1(Δn) is given by
⎧
⎪
⎪
(x−x)
2
⎪
i
⎪
−2
x [x , x),
⎪
(x x
⎪
i −i−2)(xi−1− )
∈i2i
2
−1
xi−
−
⎪
⎨
⎪
⎪
⎪

(x−xi2)(xi−x)

(xi+1−x)(x−x)−
+
i−1
x [x , x),
(x −x)(x −x ) (x− x −x
i1i
x)( )

i i−2 i i−1 i+1 i−1 i i−1
∈ −
B2,i(x) =
⎪
⎪
⎪
⎪
(1.1.1.43)
⎪
x)
2
(xi+1
⎪
−
⎪

x [xi, xi+1),
(xi+1 −xi−1 )(xi+1−xi)
∈

0 otherwise
for i = 0, 1, and
⎪
⎪
· · · , n
⎩
⎪

⎪
⎧
⎨
(x−x
2
n1)−
x x),
(xn−
n
xn
∈ [xn−1,
−1)
2
B2,n+1(x) = (1.1.1.44)

0 otherwise.
⎪
⎩

10 1.1. Singular Integral Equation
For given n ∈ N, an approximation un to the unknown u is deﬁned as
n
�
+1

un(x) = ciB2,i(x), x
i=0
∈ [0, 1] (1.1.1.45)
where ci, i = 0, 1, · · · , n + 1 are constants to be determined. For getting approximate solution un(x)
of equation

1
u(x) − λ
�
K(x, s)u(s) ds = f(x), x ∈ [0, 1] (1.1.1.46)
0
in the linear span of S2,1(Δn), one replaces u(x) by un(x) in the above. Then its evaluation at the
nodes xi, i = 0, 1, · · · , n + 1 provides a system of linear simultaneous equations

1
un(xi) − λ
�
K(xi, s)un(s) ds = f(xi), i = 0, 1,
0
· · · , n + 1 (1.1.1.47)
involving a (n + 2) × (n + 2) matrix. Here it is assumed that x xn+2 = x−2= x−1 = x0, n+1 = xn.
Solution of this system of equations provide the unknown coeﬃcients cn, whose substitution into
(1.1.1.45) gives the approximation un(x) to the solution u(x).
1.1.1.6 Method based on product integration
We consider here the numerical solution of Fredholm integral equations of the second kind with
singular kernels, for which the associated integral operator K is still compact on C(D) into C(D).
The main ideas presented here can be extended to higher dimensions, but it is more instructive to
ﬁrst present these ideas for integral equations
Mandal
NBandPanjaM
Mof a single variable,
�
b
λ u(x) − K(x, t) u(t)ds = f(x), a ≤ x ≤ b. (1.1.1.48)
a
In this setting, kernel functions K(t, s) have an inﬁnite singularity, and the most important examples
are weakly singular kernels, viz., kernel ln|t − s| with logarithmic singularity, or the kernels of the
form |t − s|
γ−1
for some 0 < γ < 1 with singularities of algebraic nature and variants of them.
We introduce the idea of product integration by considering a special case of (1.1.1.48)
�
b
λ u(x) − L(x, t) ln|x − t| u(t)dt = f(x), a x b (1.1.1.49)
a
≤ ≤
with the kernel
K(x, t) = L(x, t) ln|x − t|. (1.1.1.50)
We assume that L(t, s) is a function smooth enough (that is, it is several times continuously
diﬀerentiable), and initially we assume the unknown solution u(x) is also well-behaved.
To solve (1.1.1.49), we deﬁne a method called the product integration (trapezoidal) rule.
Let, n ≥ 1 be an integer (number of subdivisions of the domain [a, b]), h =
b−a

(length of the
n
(uniform) subintervals), and xj = a + jh, j = 0, 1, 2, ..., n (nodes). For general u ∈ C[a, b], deﬁne
1
[L(x, t) u(t)]n = [(xj − t)L(x, xj1)u(x) + (t x)L(x, x)u(x)], (1.1.1.51)
h
− j−1 −j−1 j j

11 1.1. Singular Integral Equation
for xj1 ≤ t ≤ x 1, 2, .., n and a − j , j = ≤ x ≤ b. This is piecewise (linear) polynomial in t (in case of
trapezoidal), and it interpolates L(x, t)u(t) at t = x0, x1, ..., xn, for all x ∈ [a, b].
Deﬁne a numerical approximation to the integral operator in (1.1.1.49) by
Kn u(x) =

b
[L(x, t)u(x)]n ln|x − t| dt, a ≤ x ≤ b. (1.1.1.52)
a
This can also be written as
n

Kn u(x) =
n
ωj (x)L(x, xj )u(xj ), u ∈ C[a, b] (1.1.1.53)
j=0
with weights
1
x1
ω0(x) =

(x1 − t) ln|x − t| dt, (1.1.1.54)
h
x0
1
xj
1
xj+1
ωj (x) =

(t−xj1) ln|x−t| dt+

(xj+1 −t) ln|x−t| dt, j = 1, 2, , n1, (1.1.1.55)
h
x
−
j
h
xj
· · ·
−1
−
1
xn
ωn(x) =

(t xn1) lnx t dt.
h
x
−
n1
− | − |
−
To approximate the integral equation (1.1.1.49), we use
−
n
n
λ un(x) ωj (x)L(x, xj )un(xj ) = f(x), a ≤ x ≤ b. (1.1.1.56)
j=0
Mandal
NBandPanjaM
M
As with the other methods discussed earlier, this is equivalent to ﬁrst solving the system of linear
equations
n
λ un(xi) −
n
ωj (xi)L(xi, xj )un(xj ) = f(xi), i = 0, 1, 2, ..., n (1.1.1.57)
j=0
followed by the use of the Nystr¨om interpolation formula
n

u
⎡
1
n(x) =⎣f(x) + j
⎤
n
ω (x)L(x, xj )un(xj )⎦, a
λ
≤ x ≤ b. (1.1.1.58)

j=0
With this method, we approximate those parts of the integrand in (1.1.1.49) that can be well
aproximated by piecewise (linear) polynomial interpolation, and we integrate exactly the remaining
more singular parts of the integrand.
1.1.2 Kernel with weak (logarithmic and algebraic) singularity
The conditions for existence, uniqueness and regularity of the solution of Eq. (1.1.1.1) and the
estimate of error in its approximation have been presented in the following theorems:
Assumption: It is assumed that the kernel K(x, s) is of the form
K(x, s) = g(x, s)κ(x − s) (1.1.2.1)

��
M M Panja and B N Mandal

12 1.1. Singular Integral Equation
with g to be thrice continuously diﬀerentiable function on [0, 1] × [0, 1], κ is twice continuously
diﬀerentiable function on [−1, 1] − {0} such that
c
|κ

(s)|≤ , 0 < β < 3 (1.1.2.2)
|s|
β
for every s ∈ [−1, 1] − {0}. It is important to observe that the kernel K(x, s) of Eq. (1.1.1.46)
may have weak singularity at x = s if 2 ≤ β < 3, K(x, s) is bounded but its derivative may be
singular for 0 < β < 2. Integral equations with kernels satisfying these properties often arise in the
potential theory, atmospheric physics and many other ﬁelds in applied sciences (Pallav and Pedas,
2002). Regarding the mathematical structure of the range of the operator (Id − λK)[u], we deﬁne

C
3, β
[a, b] =f ∈ C[a, b] ∩ C
3
(a, b) : |f
(3)
(x) − f
(3)
(y)|< E |x − y|
β
, x, y ∈ [a, b]. (1.1.2.3)
Here by C[a, b] we mean the Banach space of continuous functions y(t), t ∈ [a, b], with the y
C[a,b]
= max |y(t)|. C
3
(a, b) is the set of all three times continuously diﬀerentiable functions y(t), t ∈ (a, b).
a<t<b
Notice that C
3,β
[a, b], 0 < β < 3, is a Banach space with respect to the norm
|y (t)|
y = y + sup , y ∈ C
3,β
[a, b].
C
3,β
[a,b] C[a,b]
a<t<b
(t − a)
−β
+ (b − t)
−β
Theorem 1.7. Let the assumptions (1.1.2.1) and (1.1.2.2) hold and let f ∈ C
3,β
[0, 1]. Let the
homogeneous integral equation corresponding to the equation (1.1.1.1) (with f = 0) have only the
trivial solution u = 0. Finally, let the interpolation points
x
0 = 0, xi = ti−1 + η(t i − ti−1), i = 1, · · · , n, x n+1 = 1 η ∈ (0, 1) (1.1.2.4)
with the quasi-uniform grid deﬁned in (1.1.1.39) and (1.1.1.40) or the graded grid deﬁned in (1.1.1.39)
and (1.1.1.41) be used.
Then equation (1.1.1.1) has a unique solution u ∈ C[0, 1] and for all suﬃciently large n ∈ N, say
n ≥ n
0, the collocation conditions (1.1.1.25) determine a unique approximation u n ∈ S2,1(Δn) to u.
For n ≥ n
0 the following error estimate holds:
u
n − u ≤ cδ n, (1.1.2.5)
C[0,1]
where c is a positive constant not depending on n and
δ
n = n
3−β
, (1.1.2.6)
if the quasi-uniform grid deﬁned in (1.1.1.39) and (1.1.1.40) is used, and

−r(3−β ) 3
n 1 ≤ r ≤ ,
(3−β )
= (1.1.2.7)δn −3 3
n r >
(3−β )
,
if the graded grid deﬁned in (1.1.1.39) and (1.1.1.41) is used.

13 1.1. Singular Integral Equation
1.1.3 Integral equations with Cauchy singular kernel
Let us consider a singular integral equation of the form
b

1
v(t) 1
1
a v(x) − − dt + L(x, t) v(t) dt = f(x) (1.1.3.1)
ıπ
−1 x − t ıπ

−1
where L(x, t), f(x) are given H¨older continuous functions, a, b are given real or complex numbers
satisfying the conditions a
2
− b
2
= 0, b > 0, and v(x) is an unknown function. The theory of this
equation is well known and it is presented in the monographs by Muskhelishvili (Muskhelishvili,
2013), Gohberg and Krupnik (Gohberg and Krupnik, 1992), Mikhlin et al. (Mikhlin et al., 1994),
Estrada and Kanwal (Estrada and Kanwal, 2000) and others. In this section we ﬁrst present a few
methods of constructing approximate solutions of (1.1.3.1). Subsequently, a method of constructing
approximate solutions depending on the index of the characteristic equation has been discussed.
Moreover, some results on the convergence of the approximate solution have been provided.
1.1.3.1 Method based on Legendre polynomials
In an approximation scheme for getting approximate solution of Fredholm integral equation of second
kind with Cauchy singular kernel

1
u(t)
u(x) + λ dt = f(x), (1.1.3.2)
−1 x − t
a basis comprising Legendre polynomial {Pn(x), n = 0, 1, 2, ... may be used to approximate the
unknown solution
Mandal
NB
}
andPanjaM
M
n
N

u(x) ≈ cnPn(x) ≈ (P0(x), P1(x), ..., P
T
n(x)) · (c0, c1, ..., cN ). (1.1.3.3)
n=0
In their study, Abdou and Nasr (Abdou and Nasr, 2003) used (1.1.3.3) into an alternative (equiva
lent) form

1
u(y) u(x) 1
u(x) + λ
−
dy λ u(x) ln
�
− x
−
�
= f(x) (1.1.3.4)
x − y 1 + x
−1
of (1.1.3.2) and get
n
N�
1
Pi(y) Pi(x) 1 x
ci
�
Pi(x) + λ

−
dy − λ ln
−
1
�
P
+
i(x) = f(x). (1.1.3.5)
x − y 1x
i=0
−
� �
�
To reduce this equation to a system of algebraic equations, multiplication of both sides of (1.1.3.5)
by Pj (x) followed by integration over (−1, 1) provides
n
N

aji ci = fj , j = 0, 1, ..., N (1.1.3.6)
i=0

14 1.1. Singular Integral Equation
where
1
2
1
P( )
aji =
iy) Pi(x
δ

ij + λ P
j+ 1

−
(
x
j x) dy dx
2

y

−1

−1
1
−
1
−λ ln
− x
Pi(x)P(x) dx, (1.1.3.7)
1 1 +
j

x
−
1
fj = f(x)Pj (x) dx. (1.1.3.8)
−1
One can express the above equation as
A c = f (1.1.3.9)
in compact form. The integrand of the integral in the second term is multinomial in x and y, so can
be evaluated analytically. For the evaluation of the integral in third term one can expand Pi(x)Pj (x)
i
in a series in x, viz., for Pi(x) =
i
c
i l
l
x
l=0
N
i+j

Min{r,j}
Pj (x) = c¯
i+j r i+j i j
Pi(x)
rx , c¯
r = c
r l
)
−lc (1.1.3.10
r=0 l=Max
N
{0,r−i}
and use the result

1
even,
ln

1 − x

0 n is
x
n
dx =

3
1 +x

)−

2(ψ( ψ(1+
n
) 2
(1.1.3.11)
2 2
−)
n
n is odd −1
+1
Mandal
NBandPanjaM

M
in conjunction with the recurrence relation
1
ψ(1 + z) = ψ(z) + . (1.1.3.12)
z
Then fj can be evaluated eﬃciently by using the Gauss-Legendre quadrature rule
N

fi =
N
f(x
P

i )w
P
i . (1.1.3.13)
i=1
Once values of these integrals have been obtained, these values may be used in (1.1.3.9) to get the
unknown coeﬃcients
c = A
−1
f. (1.1.3.14)
Use of the coeﬃcients into (1.1.3.3) provides approximate solution of Eq. (1.1.3.2)/Eq. (1.1.3.4).
1.1.3.2 Method based on Chebyshev polynomials
In order to obtain the approximate solution of the Cauchy singular integral equation of the second
kind bounded at both ends in terms of Chebyshev polynomials of the second kind Un(x)’s, we write
u(x) in (1.1.3.2) as

N

u(x) ≈1 − x
2
N
cn Un(x). (1.1.3.15)
n=0

15 1.1. Singular Integral Equation
Use of (1.1.3.15) and the formula (Chan et al., 2003a, Eq.(36)),
�
1
1 − y
2
Un ) −1(y
dy = −π Tn(x) (1.1.3.16)
y x
−1
�
−
in (1.1.3.2), leads to
N

λ
�
cn
�
N
1 − x
2
Un(x) + µπ
�
cn Tn+1(x) = f(x). (1.1.3.17)
n=0 n=0
Here, Tn(x) is the Chebyshev polynomial of the ﬁrst kind. Equation (1.1.3.17) can be easily trans
formed to the linear algebraic equation (1.1.3.9) through the integration of both sides between -1
and 1 after multiplying Um(x) and the use of orthogonality relation (Abramowitz and Stegun, 1948,
Sec. 22.2.5)
�
1 � π
1 − y
2
Un(y)Um(y) dy = δmn. (1.1.3.18)
2
−1
The matrix elements amn and fm, m, n = 0, 1, ..., N in this case are given by

π
1
amn = λ δmn + µπ
�
Um(x) Tn+1(x) dx (1.1.3.19)
2
−1
and
m
�
1
f= f
Mandal
NBandPanjaM
M(x) Um(x) dx. (1.1.3.20)
−1
Numerical evaluation of the integral in (1.1.3.19) is straightforward since the integrand involved is
a polynomial of degree m + n + 1. The integral in fm can be evaluated by using any quadrature
formula for nontrivial f(x).
1.1.3.3 Method based on Jacobi polynomials
A method of constructing approximate solutions depending on the index of the characteristic equa
tion of Eq. (1.1.3.1) has been discussed here. The functions L(x, t), f(x) present in Eq. (1.1.3.1) are
approximated by Chebyshev polynomials, while the unknown function is approximated by Jacobi
polynomials. Before going to the approximation, we ﬁrst deﬁne a few results involved with the
solution of this equation.
Deﬁnition 1.8. We use the symbol h0 = {f : f is H¨older continuous on (−1, 1) and has integrable
singularity at the end points x = ±1} , so that
A
|f(x)|≤ , 0 < ν < 1, (1.1.3.21)
|x ± 1|
ν
A being a constant.
Deﬁnition 1.9. We denote h(−1, 1) = { f: f is H¨older continuous on (−1, 1), bounded on [0, 1] }.

1.1. Singular Integral Equation 16
One can use the transformation
Z(x)
v(x) = u(x) (1.1.3.22)
a
2
− b
2
in (1.1.3.1) to transform equation for v(x) to an equation for u(x),
1
�
1

1
BZ(t)u(t) 1 L(x, t)Z(t)u(t)
A Z(x)u(x) − dt + dt = f(x), 0 < x < 1 (1.1.3.23)
iπ
1 x
− − t iπ
�
2
−1 a
2
− b
where A =
a
2 2 , B =
b
2 2 , and Z(x) is the fundamental solution of the linear conjugate problem
a−b a −b
given by
a
X
+
− b
(x) = X
−
(x). (1.1.3.24)
a + b
To ﬁnd Z(x), we denote (Karczmarek et al., 2006)
a b θ 1
G =
−
= |G|e
iθ
, w1 = , w2 = − ln|G|. (1.1.3.25)
a + b 2π 2π
Then for 0 < θ < π,
−
�
a
2
−
2
−

Z(x) = b(1x)
α
(1 + x)
β
, α = −1 + w1 + iw2, β = −w1 − iw2, (1.1.3.26)
while for −π < θ < 0
Z(x) =
�

a
2
− b
2
(1 − x)
α
(1 + x)
β
, α = w1 + iw2, β = −1 − w1 − iw2. (1.1.3.27)
In both cases, −1 < Re(α), Re(β) < 0, so that Z(x) has integrable singularities at ±1. There are
other choices of Z(x) (fundamental solution of
Mandal
NBandPanjaM
M linear conjugate problem), viz., for 0 < θ < π,

Z(x) = a
2
− b
2
(1 − x)
α
(1 + x)
β
, α = w1, β = 1 − w1 − iw2, (1.1.3.28)
while in case −π <
�
θ < 0,

Z(x) = −
�
a
2
− b
2
(1 − x)
α
(1 +

x)
β
, α = 1 + w1 + iw2, β = −w1 − iw2. (1.1.3.29)
Here, in both cases, 0 < Re(α), Re(β) < 1. Consequently, these functions are bounded and are in
H¨older class. Whenever L(x, t) = 0, equation (1.1.3.23) becomes,

B
1
Z(t)u(t)
A Z(x)u(x) −
�
dt = f(x), 1 < x < 1. (1.1.3.30)
ıπ
1x
−
− t
−
The solution of Eq. (1.1.3.30) may be element of either h0 or h(−1, 1). The solution in h0 (i.e.,
κ = 1) is given by (Karczmarek et al., 2006)
�

1
a 1 b f(t)
u(x) = f(x) dt + C1 (1.1.3.31)
Z(x)
−
ıπ
−
1 Z(t) t − x
−
where C1 is an arbitrary constant. This arbitrary constant has been determined by demanding that
the solution satisfy an additional condition
�
1
1
BZ(t)u(t)dt = A0 (1.1.3.32)
ıπ
−1

17 1.1. Singular Integral Equation
where A0 is an arbitrary constant. Since C1 in (1.1.3.31) and A0 in (1.1.3.32) are arbitrary, the
choice C1 = A0 is admissible. This provides a unique solution to the Eqs. (1.1.3.30) and (1.1.3.32).
The necessary and suﬃcient condition for the existence of nontrivial solution in h(−1, 1) of Eq.
(1.1.3.30) is that (Karczmarek et al., 2006) the input function f(x) satisfy the condition

1
f(x)
b dx = 0. (1.1.3.33)
1 Z(x)
−
In this case, the exact solution is given by the same formula (1.1.3.31) with C1 = 0. To approximate
solution of (1.1.3.30) in the basis comprising of Jacobi
(α,β)
polynomials, P
k
(x), we use the following
results (Karczmarek et al., 2006)
1 (α,β)
(α,β) B Z(t)P (t)
A Z(x)P
k
k
(x) −
iπ

dt
⎧
−1 x − t

⎨
⎪
(α, β)

1

− −

P
k1
(x) α + β = −1, k = 1, 2, ·
=
β
· ,
2
(
−
)
·
⎪
⎩
P
−α,−
k
(x) α + β = 0, k = 0, 1, 2, · · · , (1.1.3.34)
(α, β)
2P
− −
k

+1
(x) α + β = 1, k = 0, 1, 2, · · · .
Solution of the characteristic equation
In this section, we will derive an approximate solution of Eq. (1.1.3.30) in the class h0(for the case
κ = 1). For this purpose, we will approximate the function f(x) by Chebyshev polynomials fn(x)
of degree n with Chebyshev nodes
(2k − 1)
Mandal
NBandPanja
π
xk = cos
M
M
, k = 1, 2, · · · , n + 1. (1.1.3.35)
2(n + 1)
We will use the following approximation formula (Paszkowski, 1975)
n n+1
2
fn(x) =
n n
Tj (xk)f(xk)

T

j (x), (1.1.3.36)
n+ 1
j=0k=1
where Tj (x) = cos(j cos
−1
x) are Chebyshev polynomials of the ﬁrst kind. By expressing Chebyshev
polynomials Tj (x) in terms of Jacobi polynomials, we obtain
j

Tj (x) =
n
(−α,−β)
ρjlP
l
(x), (1.1.3.37)
l=0
where,
1 1
ρ

1
(α, β)
jl = q(t)Tj (t)P
− −
(−α,−β)
π
l
(t)dt
h −1
l
1)
−α (α, β)
=
−1 1
Res (z

(z + 1)
−β
T(z)P
− −
(z), (1.1.3.38)
sinπα
−
j
(α,−β) l
h
z=
l
∞

−

q(t) = (1 −

t)
−α
(1 + t)
−β
, α + β = −1, 0 < θ < π, (1.1.3.39)
2
1
(−α,−β) 1

(−α,−β) 1 Γ(l α + 1)Γ(l β + 1)
h
l
= q(t)P
π
l
(t)

dt =
− −
. (1.1.3.40)
2lπ l Γ(l)
−1 !

18 1.1. Singular Integral Equation
Using (1.1.3.37), the interpolation polynomial (1.1.3.36) takes the form

fn(x) =
�
n
fk P
(−α,−β)
k
(x), (1.1.3.41)
k=0
where,

�
n
n+1
f0 =
2
Tj (xi)f(xi) ρj0, (1.1.3.42)
n+1
j=0
�
i

�

�
=1

n n +1
fk =
2

�
� �
Tj (xi)f(xi)
�
ρjk, k = 1, 2, · · · , n. (1.1.3.43)
n+1
j=ki=1
An approximate solution un+1(x) of the problem (1.1.3.30), (1.1.3.32) is deﬁned as a solution of the
following problem:
�
1
1
u(t)
A Z(x)un+1(x) − − BZ(t)
n+1
dt = fn(x) , −1 < x < 1 (1.1.3.44)
ıπ x
�
−t
−1
1
1
BZ(t)un+1(t)dt = A0 )
ıπ
, (1.1.3.45
−1
where fn(x) is given by (1.1.3.41). Here we have used
n
un+1(x) =
Mandal
NBandk
PanjaM
M
�
+1

(α,β
ckP
k

)

(x), (1.1.3.46)
=0
with unknown coeﬃcients {ck, k = 0, 1, · · · , n + 1}. Substitution of (1.1.3.46) in (1.1.3.44), then use
of (1.1.3.34) provides

n
�
+1

1
n

(−α,−β)
=
�
(α, β)
ckP
k1
(x) fkP
2
k

− −
(x), 1 x 1, (1.1.3.47)

−
k=0
k=1
− ≤ ≤
n
�
+1

1
1
(α,β)
ck B Z(t)P (t)dt = A . (1.1.3.48)
πi
k
�
k 0
−1
=0
(α, β)
Using the orthonormal property of P
− −
k

(x) with respect to the weight Z(x) = (1−x)
−α
(1+x)
−β
,
the coeﬃcients c1, · · · , cn+1 can be found as
c1 = 2f0, c2 = 2f1, · · · , cn+1 = 2fn.
Furthermore, using the result (Karczmarek et al., 2006)
�

1
1
(α,β) 0, k = 1, 2, ..., n + 1,
B Z(t)P
k
(t)dt =
�
(1.1.3.49)
πı
−1 1, k = 0
the rest coeﬃcient c0 can be found as
c0 = A0.

�
� �
M M Panja and B N Mandal

19 1.1. Singular Integral Equation
Next we consider the case where solution is in the class h(−1, 1) (i.e., κ = −1). In this case, the
approximate solution u
n−1 is deﬁned as the solution of the equation
�
1
1 u n−1(t)
AZ(x)u
n−1(x) − BZ(t) dt = f n(x) + γ, (1.1.3.50)
ıπ x − t
−1
where,
n−1
(α,β)
un−1(x) = c kP (x). (1.1.3.51)
k
k=0
The unknown constant γ has to be chosen in such a way that the condition
�
1
1
bZ
−1
(t)[f n(t) + γ]dt = 0 (1.1.3.52)
πı
−1
is satisﬁed. This gives (Karczmarek et al., 2006)
�
−1
γ =
1
bZ
−1
(t)fn(t)dt = −f 0. (1.1.3.53)
πi
1
Applying the formula (1.1.3.34) corresponding to α + β = 1 into (1.1.3.50) one gets
n−1 n
(−α,−β ) (−α,−β )
2 c kP (x) = f kP (x) + γ. (1.1.3.54)
k+1 k
k=0 k=0
(α,β)
Use of the orthonormal property of P gives the the coeﬃcients
k
1 1 1 1
c
0 = f 1, c1 = f 2, c3 = f 3, · · · , cn−1 = f n−1, γ = −f 0. (1.1.3.55)
2 2 2 2
Although the method of solution (determination of unknown coeﬃcients {c
k, k = 0, 1, · · · , n + 1}
in the former case and {c
k, k = 0, 1, · · · , n} in the second case) seems to indicate that the solution
obtained by this method is exact, the inaccuracies are hidden in the coeﬃcients f
i, i = 0, 1, · · · , n
(α,β)
of approximation (1.1.3.41) of the input function f(x) in the basis P (x). So, it is desirable to
k
provide an estimate of the error in the approximate solution obtained here.
Some results on estimation of error (Karczmarek et al., 2006)
Deﬁnition 1.10. Let r > 0, 0 < µ < 1. We say that a function f(x), x ∈ [−1, 1], belongs to the
class W
r
H
µ
if all the derivatives up to the order r exist and the r
th
derivative is H¨older continuous,
i.e.,
| f
(r)
(x) − f
(r)
(x
�
) |≤ C | x − x
�
|
µ
, x, x
�
∈ [−1, 1] (1.1.3.56)
where C and µ are constants independent of the choice of points x, x
�
.
Theorem 1.11. Let the input function f(x), being the right hand side of (1.1.3.1), belongs to the
class W
r
H
µ
, r ≥ 0, 0 < µ ≤ 1. Let f(x) be approximated by the interpolation polynomial (1.1.3.36)
with respect to Chebyshev nodes of the ﬁrst kind given by (1.1.3.35) and let u(x) and u
n+1(x) denote

20 1.1. Singular Integral Equation
exact and approximate solutions of the problem given by (1.1.3.30),(1.1.3.32),(1.1.3.44),(1.1.3.45).
Then the following estimation holds:

ln
2
n
I u(x) − un+1(x) I ≤ M , (1.1.3.57) ∞
n
r+µ
where M is a constant not depending on n.
Theorem 1.12. Let us suppose that the conditions of Theorem 1.11 are satisﬁed (i.e., the right-
hand side function f(x) and its approximation fn(x) are the same as in Theorem 1.11), and let
u(x), un1(x) denote exact and approximate solutions of the problem deﬁned by (1.1.3.30),(1.1.3.50) −
respectively. Then the following estimate holds:
ln
2
n
IZ(x)[u(x) − un1(x)] . (1.1.3.58) − I
∞
≤ M
n
r+µ
In this case, the integral on the right-hand side of (1.1.3.31) can have an integrable singularity at
the endpoints of the interval [-1,1]. Therefore, we have to estimate the product
1

1
1
f(t) − fn(t)
Z(x) bZ
−
(t) dt,
1 x
−
−1 < x < 1
πi t −
for which we have the following estimation

f(t)

1
1
f
2

n(t) lnn
Z(x) bZ
−1
(t)
−
dt M .
πi

t x
≤
n
r+µ
−1 −

∞
1.1.4 Integral equations with hyp
Mandal
NBandPanjaM
Mersingular kernel
Many important problems of engineering mechanics, like elasticity, plasticity, fracture mechanics
and aerodynamics can be reduced to the solution of a ﬁnite-part singular integral equation, or to a
system of such integral equations (Ladopoulos, 2000). Hence, it is of interest to solve numerically
these systems of singular integral equations of the respective boundary value problem, instead of
the problem itself. The general property of the ﬁnite-part singular integral equations, consists of
the generalization of the Cauchy singular integral equations, which have been widely investigated
during the last decades. J. Hadamard (Hadamard, 1932), was the ﬁrst who introduced the concept of
ﬁnite-part integrals, and L. Schwartz (Schwartz, 1966; Schwartz, 2001) studied some basic properties
of them. Many years later, H. R. Kutt (Kutt, 1975) proposed some algorithms for the numerical
evaluation of the ﬁnite-part singular integrals and systematically explained the diﬀerence between a
ﬁnite-part integral and a generalized principal value integral.
Sometime later M. A. Golberg (Golberg, 1983) studied the convergence of several numerical
methods for the evalution of ﬁnite-part singular integrals. The method proposed by Golberg, was
an extension beyond the Galerkin and collocation methods used for CPV integrals (Golberg, 1985).
Subsequently, A. C. Kaya and F. Erdogan (Kaya and Erdogan, 1987a; Kaya and Erdogan, 1987b)
investigated complicated problems of elasticity and fracture mechanics theory, which are reduced to
the solution of ﬁnite-part singular integral equations.
Subsequently, Ladopoulos introduced and investigated several approximation methods (Ladopou
los, 1987; Ladopoulos, 1988a; Ladopoulos, 1988b; Ladopoulos, 1989; Ladopoulos, 1994) for the nu
merical solution of the ﬁnite-part singular integral equations of the ﬁrst and the second kind. He

M
M Panja and B N Mandal

21 1.1. Singular Integral Equation
further applied this type of integral equations to the solution of several important problems of elas
ticity, fracture mechanics and aerodynamics. In his subsequent attempt Ladopoulos (Ladopoulos,
1992) introduced a generalization of the Sokhotski-Plemelj formulae, in order to show the behaviour
of the limiting values of the ﬁnite-part singular integrals. Beyond the above, E. G. Ladopoulos,
V. A. Zisis and D. Kravvaritis (Ladopoulos et al., 1988; Ladopoulos et al., 1992) used functional
analysis as a tool of investigation. They studied ﬁnite-part singular integral equations deﬁned in
general Hilbert spaces and L
p
spaces and applied them to several basic crack problems.
N. P. Vekua (Vekua, 1967) was the ﬁrst to introduce the method of regularization for the solution
of Cauchy singular integral equations. Beyond the above, the general theory of approximate methods
for solving singular integral equations was further improved by V. V. Ivanov (Ivanov, 1976) while
they used some basic topics of functional analysis.
During the last few decades, the numerical methods for getting approximate solutions of singular
integral equation with applications to several basic ﬁelds of engineering mechanics, like elastic
ity, plasticity, aerodynamics and fracture mechanics, water waves have been studied and improved
by several researchers. Survey on mathematical modelling, their exact and approximate solutions
and applications to physical problems can be found in the monographs of Ivanov (Ivanov, 1976),
Ladopoulos (Ladopoulos, 2000) and Lifanov et al. (Lifanov et al., 2004) and references therein.

M
M Panja and B N Mandal

Chapter 2
Multiresolution Analysis of
Function Spaces
In approximation theory, trigonometric or exponential functions or orthogonal polynomials associ
ated with some self-adjoint operators play the role of a basic building block in the approximation
of functions or representation of operators. If the mathematical building block consists of a ﬁnite
number of elements (say n) in the basis, operators and functions are represented by n × n matri
ces and n × 1 vectors respectively. Then the action of the operator on function or composition of
operators requires O(n
2
) or O(n
4
) number of operations. This appears to be the main diﬃculty
for developing an algorithm for numerical solution of operator equation, arising in a mathematical
model. However, for diagonal matrices, the number of arithmetic operations can be reduced to O(n).
The matrix representation of the self-adjoint operators is diagonal for some orthonormal basis. But
it is impossible to ﬁnd such orthonormal basis for operators, which are not self-adjoint, in general.
Naturally, one has to search for an orthonormal basis in which the matrix representation of the
operators, in general, is nearly diagonal or banded matrices.
Several attempts in multiple directions to resolve the issue mentioned above have been consid
ered. At the end of eighties of the last century, a comprehensive mathematical theory known as
multiresolution analysis (MRA) of function space (Grossmann and Morlet, 1984; Lemari´e-Rieusset
and Meyer, 1986; Mallat, 1989a; Mallat, 1989b) had received considerable attention in the liter
ature on both pure and applied mathematics. Belgian mathematical physicist Ingrid Daubechies
(Daubechies, 1988a; Daubechies, 1992) has invented orthogonal wavelets with compact support to
resolve the limitation mentioned above. This novel mathematical structure provides a tool in the
approximation theory which may be regarded as a (mathematical) microscope in the mathematical
analysis of elements in the space of square integrable functions.
Wavelet analysis has now become an eﬃcient tool in diﬀerent areas of science and technol
ogy. This analysis captures the local information about the signals, operators, which are more
eﬃcient than the classical Fourier analysis. An orthogonal wavelet basis can provide the joint lo
calization(Fourier and space variable) of signals or functions. The construction of diﬀerent kinds of
orthogonal wavelet bases is possible due to MRA of function space involved with the problem. The
underlying principle of MRA is as follows.

23 2.1. MRA of L
2
(R)
2.1 Multiresolution Analysis of L
2
(R)
MRA is a systematic framework to relate the underlying space and the orthonormal wavelet bases.
An MRA is deﬁned by (Daubechies, 1988a)
Deﬁnition 2.1. A multiresolution analysis of L
2
(R) (assumed to have Hilbert space structure) is
a nested sequence (V

of closed subspace of L
2
j ) (R) such that the following properties hold:
j∈Z
P1. {0} ⊂ · · · ⊂ Vj1 ⊂ Vj ⊂ V⊂
2
( ), − j+1 · · · ⊂ LR
S
P2.
2
j∈Z
Vj = L(R) and
j
,
∈Z
= {0}
P3. f(x) ∈ Vj ⇔ f(2x) ∈ V
�
j+1 ∀ j ∈ Z,
P4. ∃ a function ϕ(x) ∈ V0, such that the set, {ϕ0,k(x) = ϕ(x − k), k ∈ Z} is an unconditional
orthogonal basis of V0,
P5.

ϕ(x)dx = 1.
R
The sequence {Vj , j ∈ Z} of vector (Hilbert) spaces Vj , satisfying properties P1-P5, constitute the
MRA of L
2
(R).
2.1.1 Multiresolution generator
The function ϕ(x) of P4 mentioned above plays the key role in MRA of L
2
(R), is known as generating
function or scale function of MRA. The property P3 and ϕ(x) V0 V1 suggest that there exists
Z
Mandal

NBand
∈ ⊂
hl(l∈) such that
PanjaM
M
√ n
ϕ(x) = 2 hl ϕ(2x − l). (2.1.1.1)
l∈Z
This property of scale function is known as a two-scale relation or reﬁnement equation with mask
or low-pass ﬁlter {hl, l ∈ Z}.
The scale function and its translates at higher resolution j are deﬁned as ϕ
j
j,k(x) = 2
/2
ϕ(2
j
x−k).
2.1.2 Wavelets
The additional information in the approximation subspace Vj+1 compared to the approximation
space Vj , is hidden in Wj = Vj+1 −Vj , the orthogonal complements of Vj in Vj+1. The assumption,
Hilbert space structure of L
2
(R) and all its subsets Vj , j ∈ Z provide the orthogonal decomposition

Vj+1 = Vj Wj . (2.1.2.1)
The nested property of the subspaces Vj in P1 gives
�

⊥ ∀

Wj
! Wj
!! j
'
= j
''
≥ j, (2.1.2.2)
and
J−1

VJ = Vj0
�
Wj , for j0 < J. (2.1.2.3)
j=j0

24 2.1. MRA of L
2
(R)
Finally, P2 of MRA ensures that

L
2
(R) =
�
Wj . (2.1.2.4)
j∈Z
Each Wj is a Hilbert subspace and provides an option for the existence of an orthonormal basis.
The space Wj is known as detail space and the element ψ (in W0) is called the mother wavelet.
From the property (2.1.2.1), for j = 0 it appears that ψ(x) can be expanded using the basis of
V1 as
�
ψ(x) =
√
2 gl ϕ(2x − l). (2.1.2.5)
l∈Z
This property creates a connection between diﬀerent scales of approximation space and detail space.
It is important to mention here that the coeﬃcients gl’s involved in (2.1.2.5) are known as high-pass
ﬁlter of MRA generated by the scale function ϕ(x). These two sets of coeﬃcients (viz., hl, l ∈ Z,
gl, l ∈ Z) play the key role in the development of wavelet based numerical scheme, and will be
discussed in somewhat detail in the subsequent chapters. The wavelet and its translates at higher
resolution j are deﬁned as ψ = 2
j/2
ψ(2
j
j,k(x) x−k). The celebrated work of Daubechies (Daubechies,
1988a) provided explicit construction of ﬁnite sequences of coeﬃcients {hl, l ∈ Λϕ} and generators
ϕ(x) involved in the reﬁnement equation (2.1.1.1) that have compact support, translates are or
thonormal and have varying degrees of smoothness. Here Λϕ denotes the index set corresponding
to the non-zero coeﬃcients (low-pass ﬁlter of MRA) in the two scale relation (2.1.1.1) for ϕ.
2.1.3 Basis with compact support
The underlying elements of MRA are the scale function
Mandal
N ϕ(x), its two-scale relation
B
(2.1.1.1) involving
andPanjaM
and
M
low-pass ﬁlter hk’s the relation (2.1.2.5) between wavelet ψ, translates of scale functions at
scale 2 involving high-pass ﬁlter gk’s. Most of the mathematical results and technical formulas of
MRA depend on the support of ϕ(x) as well as non-zero hk’s and gk’s. Consequently, the exercise of
technical formulae involving low- and high-pass ﬁlters, appearing in MRA of function space, becomes
easier if they are ﬁnite in numbers, i.e., scale functions or wavelets have compact support. Haar basis
(Haar, 1911) is the ﬁrst classical example of the compactly supported (not continuous) orthonormal
wavelet in L
2
(R). It is deﬁned by
�

1, 0
1
1, 0
≤ x <
≤ x < 1
⎨
⎧
2
φ(x) = , ψ(x) =
0
−
1
⎩
1,

2
≤ x < 1 (2.1.3.1)
, elsewhere
0, elsewhere.

�

j
For higher resolution,hj,k(x) = 22h(2
j
x − k), j, k ∈ Z forms a compactly supported orthonor
mal basis of Vj . But due to the poor regularity of this wa
�
velet (wavelets have jump discontinuity),
one desires a wavelet family with higher regularity. At the end of eighties of last century, Daubechies
(Daubechies, 1988b; Daubechies, 1992) ﬁrst developed generators and wavelets with compact sup
port for MRA of L
2
(R), whose regularity increases with the increase in the length of support of scale
functions and wavelets. Now, this type of wavelet is known as wavelet basis in Daubechies family.

�
� �
�
� � � � �
M
M Panja and B N Mandal

25 2.1. MRA of L
2
(R)
2.1.4 Properties of elements in Daubechies family
The important property of Daubechies wavelet is vanishing moment conditions. If the support
width of Daubechies-K (DauK) family scale function ϕ(x) and wavelets ψ(x) is 2K − 1, then the
corresponding wavelet ψ(x) has K vanishing moments, viz.,
�
∞
m
x ψ(x) dx = 0 for m = 0, 1, ..., K − 1. (2.1.4.1)
−∞
Also, it can be shown that the polynomials of degree n (0 ≤ n ≤ K − 1), can be exactly reproduced
by a linear combination of the integer shift of scale-function ϕ(x) inspite of these functions not being
square integrable in R. This aspect plays a crucial role in the approximation theory, e.g., sparseness
in the matrix representation of operators in the wavelet basis, estimation of a posteriori error in
the approximation of function, in particular. The scale functions in DauK-family with support
[−K + 1, K] at resolution 0 follow the two-scale relation
K
ϕj,k(x) = hl ϕj+1,2k+l(x) = h.Φj+1 2k(x) (2.1.4.2)
l=−K+1
�
with
h = (h−K+1, · · · · · · , hK )2K×1, Φj+1,2k(x) = (ϕj+1 2k−K+1(x), · · · · · · , ϕj+1 2k+K (x))1×2K .
The condition on ϕ(x) given in P5 suggests the elements {hl, l = −K + 1, ..., K} mentioned above
K
satisfy
√
hl = 2.
−K+1
k ∈ Z provides further condi-Furthermore, the orthogonality among the integer translates ϕk(x),
tions
K K
hlhk = δk l.
l=−K+1 k=−K+1
The relation among ψj,k(x) and ϕj,k(x) is
K
(2.1.4.3)ψj,k(x) = gl ϕj+1,2k+l(x) = g.Φj+1 2k(x)
l=−K+1
where g = (g−K+1, · · · · · · , gK )2K×1 and gl = (−1)
l
h1−l is the high-pass ﬁlter of MRA generated by
the scale function ϕ(x) in Daubechies family. Orthonormal condition among the integer translates
of ψ(x) and their orthogonality with the integer translates of ϕ(x) provides relation among elements
of low- and high-pass ﬁlters
K K K K K
glhk = 0, glgk = δk l, g l = 0.
l=−K+1 k=−K+1 l=−K+1 k=−K+1 −K+1
The set of functions {ϕj,k(x), k ∈ Z} and {ψj
�
,k(x), k ∈ Z, j
�
≥ j} are the orthonormal bases for
the approximation space Vj and detail space Wj
� at resolutions j and j
�
of the MRA of L
2
(R). The
explicit forms of low-pass and high-pass ﬁlters (Daubechies, 1992) for K = 2, 3, 6 respectively are
given in Table 2.1. The ﬁgures of scale functions and wavelets for K = 2, 4, 10 are presented in Figs.
2.1–2.2.

M M P anja and B N Mandal
2.1. MRA ofL
2
(R) 26
Table 2.1: The values of low-pass ltershl; l=K+ 1; :::; KforK= 2;3;6,g= (1)
k
k h1k.
lK= 2 K= 3 K= 6
5
9819491
124500312
4
34209766
97811521
3
71158349
133974916
2
pp p
1+ 10+ 5+2 10
p
16 2
27346967
122678536
1
p
1+ 3
p
4 2
p p p
5+ 10+3 5+2 10
p
16 2
p

19737243
123362935
0
p
3+ 3
p
4 2
p p
5 10+ 5+2 10
p
8 2

9161454
99842531
1
p
3 3
p
4 2
p p p
5 10 5+2 10
p
8 2
3244175
47055187
2
p
1 3
p
4 2
p p p
5+ 103 5+2 10
p
16 2
4060626
208648055
3
p p p
1+ 10 5+2 10
p
16 2

4111703
184118134
4
23497
59998635
5
514970
152446787
6
276333
362752699-2 -1 1 2 3 4
x
0.5
1.0
DauφK2 -4 -2 2 4
x
0.5
1.0
DauφK4 -10 -5 5 10
x
-0.4
-0.2
0.2
0.4
0.6
0.8
1.0
DauφK10
Figure2.1:Plotsofscalefunctions(a)DauK2,(b)DauK4and(c)DauK10.Regionswithnegligible
valuesofthescalefunctionhavebeenomitted.

M
M Panja and B N Mandal

27 2.2. MRA in L
2
(a, b) -2 -1 1 2 3 4
x
-1.5
-1.0
-0.5
0.5
1.0
1.5
DauψK2 -4 -2 2 4
x
-1.0
-0.5
0.5
1.0
DauψK4 -10 -5 5 10
x
-1.0
-0.5
0.5
DauψK10
Figure 2.2: Plots of mother wavelets (a) DauK2, (b) DauK4 and (c) DauK10. Regions with negligible
values of the mother wavelet have been omitted.
From these ﬁgures it appears that the scale functions are highly asymmetric. This feature creates
some problem in the development of numerical scheme based on the basis generated by ϕ and ψ. To
avoid such diﬃculties, another set of generators known as symlet have been developed. Scale func
tions in these family are not symmetric, but their asymmetry are somewhat less. Numerical values
of low-pass and high-pass ﬁlters for K = 4, 6, 10 respectively are given in Table 2.2 (Daubechies,
1992). The ﬁgures of scale functions and wavelets for K = 4, 6, 10 are presented in Figs. 2.3–2.4.
2.1.5 Limitation of scale functions and wavelets in Daubechies family
Daubechies K-family wavelet is an outstanding construction of orthonormal wavelet bases with
compact support of L
2
(R). However, some special attention for MRA of L
2
[a, b] is required. Wavelets
on the interval have great importance in numerical analysis and signal processing. As an example,
Haar basis forms an orthonormal basis of L
2
(R) as well as of L
2
[a, b]. But for the approximation of
elements in the space of continuous functions, Haar basis is not suitable due to poor regularity of
its wavelets.
2.2 Multiresolution Analysis of L
2
([a, b] ⊂ R)
If we choose Daubechies K-family wavelets having the prop
S
erty of increasing smoothness with in

crease of support width, we take the basis {ϕj0 ,k, k ∈ Z} {ψj,k, j ≥ j0, k ∈ Z} on the interval in

28 2.2. MRA in L
2
(a, b)
Table 2.2: The values of low-pass ﬁlters hl, l = −K + 1, ..., K for K = 4, 6, 10. for symlet.
k 4 6 10
−9
33511
61534905
−8
16891
249783689
−7 −
548727
89803297
−6 −
91148
87965245
−5
450843
41390792
1968189
60605419
−4
223253
90447855
369988
45068633
−3 −
1671118
31192443
−
10420351
124896922
−
5677432
50340999
−2 −
3566747
170205911
−
1658487
48548338
−
5546522
110664607
−1
6665909
18944263
108765165
313237573
39675409
118954021
0
48245878
84890737
70956571
127402874
46309010
85107181
1
37905564
179973677
15459215
64695851
35476044
130711841
2 −
5196371
74065835
−
28660593
558006355
−
1179026
46920301
3 −
464495
52118124
−
1377514
92501041
−
2029073
89701077
4
1111162
48766889
7758763
245334197
4331981
122539248
5
M
M Panja and B N Mandal

157434
125951125
151235
37100061
6 −
826173
149779355
−
1286276
89367445
7 −
31248
54939833
8
51304
15796227
9
3081
76393604
10 −
178456
549442043
such a way that the support of those bases lie within [a, b]. However, it is impossible to incorporate
the basis within [a, b] as the support of a few elements in the basis do not overlap completely in
[a, b] and also orthogonality property has been lost for some elements in the bases as observed in the
following subsection.
2.2.1 Truncated scale functions and wavelets
In their investigation, Jia et al. (Jia et al., 2003) observed that orthogonality is no longer a signiﬁcant
issue for wavelet bases of Sobolev spaces. Instead, the size of the support of wavelet turns out to
be an important criterion for its performance. From a numerical point of view, a wavelet with

29 2.2. MRA in L
2
(a, b) -4 -2 2 4
x
-0.2
0.2
0.4
0.6
0.8
1.0
1.2
SymφK4 -6 -4 -2 2 4 6
x
-0.2
0.2
0.4
0.6
0.8
1.0
1.2
SymφK6 -10 -5 5 10
x
-0.2
0.2
0.4
0.6
0.8
1.0
SymφK10
Figure 2.3: Plots of scale functions (a) SymK4, (b) SymK6 and (c) SymK10. Regions with negligible
values of the scale function have been omitted.
M
M Panja and B N Mandal
-4 -2 2 4
x
-1.0
-0.5
0.5
1.0
1.5
SymψK4 -6 -4 -2 2 4 6
x
-1.0
-0.5
0.5
1.0
1.5
SymψK6 -10 -5 5 10
x
-1.0
-0.5
0.5
1.0
SymψK10
Figure 2.4: Plots of mother wavelets (a) SymK4, (b) SymK6 and (c) SymK10. Regions with
negligible values of the mother wavelet have been omitted.

M
M Panja and B N Mandal

30 2.2. MRA in L
2
(a, b)
smaller support usually generates more eﬃcient algorithms for wavelet transforms than that with a
larger support. They exercised this principle over functions on the Sobolev space deﬁned over the
entire real line R. It is found that the above observation of Jia et al. holds good for approximating
functions deﬁned over a ﬁnite domain. Very accurate numerical estimates of deﬁnite integrals have
been obtained with a scale function based quadrature rule which depends on the set of independent
but non-orthogonal scale functions (Meyer, 1991; Goswami and Chan, 2011).
We consider here collections of scale functions ϕ(x) with support of ϕ as (α, β), α < 0 < β, α, β ∈
Z. We divide the translates of ϕ(x) at a particular resolution j into three classes (Lee and Kassim,
Λ
V IT
Λ
V RT
2006) Λ
V LT
j
,
j
,
j
given by

Φ
LT
j ϕ
LT
j l (x) = ϕj l(x)χ
[a,b](x), l ∈ {2
j
a − β + 1, · · · , 2
j

a − α − 1}= ,
1×(β−α−1)

Φ
IT
j ϕj l(x), l ∈ {2
j
a − α, · · · , 2
j
b − β} (2.2.1.1)= ,
1×(2
j
(b−a)−β+α+1)

Φ
RT
j ϕ
RT
j r (x) = ϕj r(x)χ
[a,b](x), r ∈ {2
j
b − β + 1, · · · , 2
j
b − α − 1}= .
1×(β−α−1)

Ψ
LT
j ψ
LT
j l (x) = ψj l(x)χ
[a,b](x),

l ∈ {2
j
ψj l(x), l ∈ {2
j
a − α, · · · , 2
j
b − β}
a − β + 1, · · · , 2
j
a − α − 1}= ,
1×(β−α−1)

Ψ
IT
j (2.2.1.2)=
1×(2
j
(b−a)−β+α+1)
,

Ψ
RT
j ψ
RT
j r (x) = ψj r(x)χ
[a,b](x), r ∈ {2
j
b − β + 1, · · · , 2
j
b − α − 1}= .
1×(β−α−1)
The symbol χ
[a,b](x) represents the characteristic function which has the value 1 when x ∈ [a, b] and
0 elsewhere. The superscript LT (RT ) has been used to represent elements of Φ
LT
(Φ
RT
) containing
right (left) tail of suppϕ which overlaps with left (right) end of the domain [a, b]. Elements of Φ
LT
and Φ
RT
are not orthonormal, in general. Instead, elements ϕ
LT
(x) and ϕ
RT
(x) are independent.
l r
In case of the domain R
+
(R
−
), truncated elements in the basis at resolution j = 0 are Φ
LT
=
{ϕl(x)χ
[0,∞)(x), l = −β + 1, · · · , −α − 1} (Φ
RT
= {ϕr(x)χ
(−∞,0](x), r = −β + 1, · · · , −α − 1} ).
In contrast to the two-scale relation (2.1.1.1) for interior scale functions, and single relation
among mother wavelet and integer translates of ϕ(2x), the two-scale relation among elements in the
, Φ
RT
bases Φ
LT
are diﬀerent. Those are
⎛
.
.
h−β+1 h−β 0 0 · · · · · · 0 0 0 0 .
.
h−β+3 h−β+2 h−β+1 h−β · · · · · · 0 0 0 0
.
.
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
.
.
0 0 0 0 · · · · · · hα hα+1 hα+2 hα+3 .
⎡ ⎤
ϕ
LT
−β+1
(·)
√
= 2
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
ϕ
LT
−β+2
(·)
.
.
.
.
.
.
ϕ
LT
−α−2(·)
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
ϕ
LT
−α−1(·)
.
.
0 0 0 0 · · · · · · 0 0 hα hα+1 .

⎡
ϕ
LT
β+1
(2·)
⎢
−
ϕ
LT
−β+2
(2·)
.
.
⎤
.
.
.
. 0 0 0 0
⎥
.
· · · · · ·0 0 0 0
⎢ ⎥
.
⎥
. 0 0 0 0
⎞
⎢
⎢
.
.
.
⎥
. .. .
· · ·
.
· · ·0 0 0 0

.
⎢
ϕ
LT
−α−2(2·)
. LT
⎥
.

.

. .
.
. .
⎥
.
.
. . . . .
⎟
⎟
⎢
⎢
⎟
⎟
⎢
⎢
⎟⎢
⎢
ϕ (2
−α−1·)
⎥
. .
· ·
⎥
. . .. .
.
· · · · · · ·
.
.

. .
.
.
.
.
.
.
. . . ϕα(2 )
⎥
⎥
⎥
.
⎢
−·
ϕ (2
.
⎟
.
α+1)
h h
⎟⎢ ⎥
⎥
, (2.2.1.3)
· · · · · · · · · · β−1 β0 0
⎟
−

⎢
.
·
.
⎥
.
.
⎟⎢
.
.
⎟⎢ ⎥
⎥
· · · · · · · · · · · · hβ h−1 β
⎟
⎟
⎠⎢
⎢
⎢
⎢
⎢
⎢
⎢
.
.
⎥
⎥
⎢ .

β α (2−
⎥
⎥
⎥
⎥
⎣
ϕ−2 3·)
ϕβ−2α−2(2·)
⎥

⎦
⎥

⎡
.

RT
⎛
⎤
⎜
hα h
ϕ
+1
β+1
( )
α · · · · · · · · · · · ·
.
.
⎢
−
−β+2
(·)
.
⎜ .
⎢
ϕ
RT
·
⎥
⎥
⎜
⎜
00 hα hα+1
.
·
.
· · · · · ·
. .
·
. .
· ·
.
.
.
.
⎥
⎥
⎢
⎢
⎢ .
⎥
√ . .
.
.
.
.
.
.

.
.
.
. . .
⎢
⎢
.
⎢
.
.
⎜
⎜
⎥
⎥
= 2
⎜
⎜
. . . . .
⎥ ⎜
⎜
. .
.

.

. .
. .
. . .
. . .
⎣
ϕ
RT
α
⎦
2(·)
−
ϕ
RT
−
−α1(·)
−
⎜
⎜
. .
.
⎝
0 0 0 0 · · · · · ·
.
0 0 0 0 .
.
· · ·
.
0 0 0 0 · · ·0 0 0 0 .
Mandal
NBandPanjaM ⎡
M
ϕβ α+2(2−+2 )

·
ϕ
.
−β+2α+3(2 )
.
·
.hβ1 hβ 0 0− · 0 0 0 0 .
.
⎤
· · · · ·
.
.
.
.hβ3 h− β−2 hβ−1 hβ 0 0 0
⎞
.
· 0
⎢
· ·
.
· · ·
. .
. .
⎢
⎢
.
.
⎥
.
⎥
. . .
. .

.

.

.
⎥
. . . .. . .
⎟
⎟ ⎢
⎢
ϕ (2
. . .
·)
⎥
−β
⎥
. (2.2.1.4)
. . ..
.
⎢

. . .
⎟
.
. . .
⎥
. . . . . .
⎟
⎟
⎟
⎟ ⎢
⎢
⎢
⎢
⎟
⎟
⎢
⎢· · · · · · · · ·
.
ϕ
RT .
. ).
.
⎥
⎥
⎥
(2
. 0 0 0 0 · · · · · · h h
β
α h
+1
α+1
⎥
α+2 hα+3
−
.
·
.
⎥
.
. .
⎥
. 0 0 0 0 · · · · · ·0 0 hα h
⎟
α+1
⎟
⎟
⎢
⎢
⎠
⎢

ϕ
RT
(2
⎥
⎥
⎥
⎢
⎢
α2·)
ϕ
⎥
⎢
⎣
R
−
T
−
(2 )
−α−1·
⎥
⎥
The relation among wavelets and scale functions in having partial support can be found
⎦
as
⎛
⎡ ⎤
.
g 0 0 0 0 .
(
⎜
⎜
.
g 0 0
ψ
LT
β+1
·)
−β+1 −β · · · · · ·
⎢
−
ψ
LT
(· ⎥ ⎜
.
.
) g−β+3 g−β+2 g−β+1 gβ· · · · · ·0 0 0 .−
⎢
⎢
⎢
−β+2 0
. . . . . . . .
⎢
. √ .

.

.

.

.

.
.
.
. . . . . . .

⎢
⎢
= 2
. . .
.
.
⎥ ⎜
⎜
⎥
⎥
⎥
⎜
⎦
⎥
⎥
⎥
⎜
⎜
⎜
⎜ . . . . .

. .
⎜
. .
⎣
⎢
. . .
. .
. .
. . .

ψ
LT
( ) .
−α−2·
0 0 0 0
.

g g g g .
ψ
LT
( )
· · · · · ·α α +1 α+2 α+3
−α−1·
⎝
⎜
.
·
.
0 0 0 0 · · · · ·0 0 gα gα+1.
31 2.2. MRA in L
2
(a, b)

M
M Panja and B N Mandal

⎤
ϕ
LT
−β+1
(2·)
ϕ
LT
−β+2
(2·)
⎡
(2.2.1.5)
. (2.2.1.6)
(2.2.1.7a)
⎤
.
.
.000
⎡
0
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
.
.gα gα+1 · · · · · · · · · · · ·
.
.
.gα gα+1 · · · · · · · · · ·
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
.
.
.
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0000· · · · · ·0000
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
,
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
.
.
.
.
.
ϕ
LT
−α−2(2·)
ϕ
LT
−α−1(2·)
· · · · · · · · ·
ϕ−α(2·)
ϕ−α+1(2·)
.
.
.
.
.
.
ϕβ−2α−3(2·)
ϕβ−2α−2(2·)
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
ϕ−β+2α+2(2·)
ϕ−β+2α+3(2·)
.
.
.
.
.
.
ϕ−β (2·)
.
.
.
· · · · · · · · ·
ϕ
RT
−β+1
(2·)
.
.
.
ϕ
RT
−α−2(2·)
ϕ
RT
−α−1(2·)
⎞
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
0000· · · · · ·00gβ−1 gβ
.
.
.
0000gβ−3 gβ−2 gβ−1 gβ · · · · · ·
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
· · · · · · gα gα+1 gα+2 gα+30000
.
.
.
gα gα+100· · · · · ·0000
.
.
.
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
0000· · · · · ·0000
.
.
.
0000· · · · · ·0000
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
gβ−1 gβ
· · · · · ·0000
⎤⎡
,
⎥
⎦
Φ
LT
T
(2·)
j
· · · · · · · · ·
⎢
⎣2 [H
LT
, H
LT I
]
√
=
T
Φ
LT
(·)
j
⎤
Φ
LT I
T
(2·)
j
Φ
RT I
T
(2·)
j
⎡
· · · · · · · · · · gβ−1 gβ
· · · · · · · · · · · ·
.
.
.
.
.
.
00
⎛
⎤
ψ
RT
−β+1
(·)
⎡
√
2=
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
ψ
RT
−β+2
(·)
.
.
.
.
.
.
ψ
RT
−α−2(·)
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
ψ
RT
−α−1(·)
We write these relations as
(2.2.1.7b) ,
⎥
⎦· · · · · · · · ·
⎢
⎣2 [H
RT I
, H
RT
]
Φ
RT
T
(2·)
j
√
=
T
Φ
RT
(·)
j
32 2.2. MRA in L
2
(a, b)

33 2.2. MRA in L
2
(a, b)

Φ

T

LT
√ j
(2 )

T
LT
·
LT LT I
Ψ
j , ]
⎢
·
( ) = 2 [ G G
⎡
⎣

· · ·
Φ
LT
· ·
I
·

· ·

·

T
⎤
j
(2·)

T
⎦
⎥
, (2.2.1.8a)
⎡

Φ
RT I
√
⎢
j
(2·)

T
RT
·
RT
( ) ,
T
Ψ
I
2 [G G
R
j = ] ⎣

· · · · · · · · ·

. (2.2.1.8b)

·
T
Φ
RT
⎤
j
(2 )
⎥
These relations will be useful to obtain representation of functions in H¨older
⎦
class/singular operators
non-smooth or unbounded at the boundaries of the ﬁnite interval [a, b]. It is pointed out earlier that
the elements
LT /RT LT /RT
of Φ
j
or Ψ
j
are not orthogonal. So it is
�
desirable to
�
get values of integrals of
T

LL(RR)

LT (RT )

LT (RT )
products of those elements. If one denotesN =
+( ) Φ
j
(x)Φ
j
(x) dx, then
R
−
use of the two-scale relations (2.2.1.7) will provide a system of algebraic equations for elements in
N
LL(RR)
as
LL
N =

T
LT

LT
Φ(x)Φ(x) dx
R
+

LT
T

= 2[H , H
LT I
]

Φ
j
R
+

LT
(2x)

Φ
LT
j (2x) dx
�

T
LT

H

H
LT
T.
I

�
This relation can be recast into the form

LL
= [H
LT
, H
LT
N
I
]
�
T
N
� �

O
�

LL

H
LT

T. (2.2.1.9)
O Id H
LT I
Mandal
NBandPanjaM
Following similar steps the elements at the other
M
end satisfy the equation
�

I
�
�

T
d O H
RT I

N
RR
[H
RT I
, H
RT
= ]
RR

T
�
. (2.2.1.10)
ON H
RT
Although basis Φ
LT
j
∪ Φ
I
∪ Φ
RT
j j
comprising truncated and inte

rior scale

functions in Daubechies
family successfully represents functions in H¨older class with reasonable accuracy, estimation of a
posteriori error in the approximation is not straightforward due to loss of orthogonality of elements
with truncated domains in the basis. This limitation may be avoided either by considering a new
class of basis, known as multiwavelets or a new formulation of boundary elements of scale functions
and wavelets as discussed in the following two sections.
2.2.2 Multiwavelets
Elements of multiwavelet family diﬀer from wavelets in Daubechies family in the sense that a set of
functions ϕ
0
, ϕ
1
, ..., ϕ
K−1
plays the role of scale function instead of single function ϕ. Alpert and his
coworkers (Alpert, 1993; Alpert et al., 1993; Alpert et al., 2002) are pioneers in the development of
multiwavelets involving polynomials. In the framework of MRA of L
2
([0, 1]) based on multiwavelets,
the scaling functions ϕ
0
, ϕ
1
, ..., ϕ
K−1
are dilated, translated and normalized polynomials of K com
ponents, given by
ϕ
i
(x) := Ni Pi(2x − 1), i = 0, 1, ..., K − 1; 0 ≤ x < 1, (2.2.2.1)

34 2.2. MRA in L
2
(a, b)
where Pi(x)’s are some classical orthogonal polynomials of degree i (i = 0, 1, ..., K − 1). The co

eﬃcient Ni is the normalization constant given by Ni =
√
2i + 1 for scale functions in Legendre
multiwavelets.
At resolution j these are expressed as

ϕ
i
2
j
2 ϕ
i
(2
j
j,k(x) := x − k), j ∈ N ∪ 0, (2.2.2.2)
where supp ϕ
i k k+1 i
j,k
(x

) =
j ,
j. For a given j > 0, shifting or translation of ϕ
2 2 j,k
(x) is represented
by the symbol k k =

0, 1, ..., 2

j
− 1

. In a particular resolution j, ϕ
i1
(x) is orthogonal to ϕ
i2
j,k
(x)
1 j,k2
for i1=i2 (due to orthogonality
y
of polynomials), k1=k2 (due to disjoint support) with respect

to the product <
1
inner f, g >= f(x)g(x)dx. Apart from the usual recurrence relation (in n) for
0
Pn(x)(viz. pnPn+1(x) + qnx Pn(x) + rn Pn1(x) = 0), the reﬁnement equations or the two-scale −
relations among the scale functions ϕ
i
j,k
(x) are (Alpert et al., 2002; Paul et al., 2016a)
K− K1 1
1
K
1
i
√

(0) (1)
−
r

r
1
√
K K
(s)
ϕ
j,k(x) = h
i,r
ϕ
j+1,2k(x) +h
i,r
ϕ
r
j+1,2k+1(x) = h
i,r
ϕ
j+1,2k+s(x). (2.2.2.3)
2 2
r=0 r=0 s=0

(s)
.

The elements
.
h
i,r
(s = 0, 1) of the low-pass ﬁlter H = √
1

�
h
(0)
. h
(1)
�
are determined uniquely
2
by using the deﬁnition (2.2.2.1) into the two-scale relations (2.2.2.3).
The
j
elementsψ
i
(x) := 2 ψ
i
(2
j
2
j,k
x − k) of multiwavelets ψj,k having K components for each j
and admissible k are give

n by

i
1
K−1
√
K K1 1
(0) r

(1) r
Mandal 1
−
r
ψ( )
NB
andPanjaM
M (s)
j,kx= g

i,r
ϕ
j+1,2k(x) +g
i,r
ϕ
j+1,2k+1(x)

= √
K K
g

i,r
ϕ
j+1,2k+s(x). (2.2.2.4)
2 2
r=0 r=0 s=0

(s)
.

The elements g
i,r
(s = 0, 1) of
.
the high-pass ﬁlter G = √
1
g
(0)
. g
(1)
are obtained by using the
2
following relations at resolution 0 (Alpert, 1993; Alpert et al.,
�
2002) :
�
y
1
ψ
i
0,0(x)x
m
dx = 0 for i = 0, 1, ..., K − 1; m = 0, 1, ..., K − 1 + i,
0
y
(2.2.2.5)

1
ψ
i1
(x)ψ
i2
0,0 0 ,0
(x)dx = δi1,i2 for i1, i2 = 0, 1, ..., K
0
− 1.
Explicit values of the elements of h
(0)
, g
(0)
for K = 4 are
⎡
1 0 0 0
⎤
⎢
⎡
0 √
2
85

12
17

21
⎢ ⎥
⎥
−
85
⎢
√
3 1
⎢
−
2
0 0
2 ⎥
⎥
⎢
⎢
⎢
− √
1
−√
1
(0)
, g
(0)
=
21 7
−
h=

5
√
3
⎢
84 2
⎤
⎢
(2.2.2.6)
√
.
⎢ −
15
0
1
0

4 4
⎥
√ √ √
⎥
⎥
⎥
7
⎥
⎣
21
−
35 1
8 8 8 8
(1)
⎥
⎦
⎢
⎣
63 8
0 −

21
340
−

272
−√
85

125
√
125
1344 448 8
√
23 15
21 8
⎦
⎥

The elements gof g
(1)
can be found from the elements of g
(0)
i,j
by using the formulae (cf.
equation(3.27) in (Alpert et al., 2002))
(1) i+j+K (0)
g
i,j
= (−1) g
i,j
. (2.2.2.7)

�
�
�
�
�
�
M
M Panja and B N Mandal

Closed form expressions of scale functions and wavelets for the LMW family with K = 4 are given
by Lakestani et al. (Lakestani et al., 2011). For K = 5 the closed form expressions of scale functions
are given by
ϕ
0
(x) = 1, 0 ≤ x < 1,
√
ϕ
1
(x) = 3(2x − 1), 0 ≤ x < 1,
√
ϕ
2
(x) = 5(6x
2
− 6x + 1), 0 ≤ x < 1,
√
ϕ
3
(x) = 7(20x
3
− 30x
2
+ 12x − 1), 0 ≤ x < 1,
√
ϕ
4
(x) = 9(70x
4
− 140x
3
+ 90x
2
− 20x + 1), 0 ≤ x < 1,
√
ϕ
5
(x) = 11(252x
5
− 630x
4
+ 560x
3
− 210x
2
+ 30x − 1), 0 ≤ x < 1.
(2.2.2.8a)
The closed form expressions of wavelets for K = 5 are given by
ψ
0
(x) =
ψ
1
(x) =
⎧
⎨
⎩
⎧
⎨
⎩
⎧
⎪
⎪
⎨
√
1 4 1
−31 + 900x − 5880x
2
+ 13440x
3
− 10080x 0 ≤ x <
2
,
93
√
1 4 1
1651 − 10860x + 26040x
2
− 26880x
3
+ 10080x ≤ x < 1,
293
√
1 4 1
13 − 486x + 3990x
2
− 11200x
3
+ 10080x 0 ≤ x <
2
,
19
√
1 4 1
2397 − 14214x + 30870x
2
− 29120x
3
+ 10080x ≤ x < 1,
219
35 4 1
7347
−31 + 1386x − 13296x
2
+ 42720x
3
− 43200x 0 ≤ x <
2
,
ψ
2
(x) =
⎪
⎪
⎩
⎧
⎪
⎪
⎨
35 4 1
12421 − 69846x + 144336x
2
− 130080x
3
+ 43200x ≤ x < 1,
7347 2
21 4 1
19
−1 + 52x − 570x
2
+ 2060x
3
− 2310x 0 ≤ x <
2
,
ψ
3
(x) =
⎪
⎪
⎩
⎧
⎪
⎪
⎨
21 4 1
−769 + 4148x − 8250x
2
+ 7180x
3
− 2310x ≤ x < 1,
19 2
7 4 1
79
−1 + 60x − 750x
2
+ 3060x
3
− 3840x 0 ≤ x <
2
,
ψ
4
(x) =
⎪
⎪
⎩ 7 4 1
79
1471 − 7620x + 14610x
2
− 12300x
3
+ 3840x
2
≤ x < 1.
(2.2.2.8b)
Values of low- and high-pass ﬁlters, as well as explicit expressions for their wavelets for K =
2, 3, · · · , 10 for several families of multiwavelets have been presented in Appendix A and Appendix
35 2.2. MRA in L
2
(a, b)

2.2. MRA inL
2
(a; b) 360.2 0.4 0.6 0.8 1.0
x
-1
1
2 0.2 0.4 0.6 0.8 1.0
x
-3
-2
-1
1
2
3
4
a) b)
Figure2.5:PlotsofLegendremultiscalefunctions'
i
(x)incaseofK=6:a)'
0;1;2
(x),b)'
3;4;5
(x).0.2 0.4 0.6 0.8 1.0
x
-2
2
4 0.2 0.4 0.6 0.8 1.0
x
-2
2
4
a) b)
Figure2.6:PlotsofLMW
i
(x)withK=6a)
0;1;2
(x),b)
3;4;5
(x).
B.Componentsofscalefunctions(K=6)and
MandalNBand PanjaM M
waveletsinLMWbasisforK=6areillustrated
graphicallyinFig.2.5andFig.2.6respectively.Itmaybenotedthat
0
(x);
2
(x)and
4
(x)are
discontinuousatx=
1
.TheseareillustratedgraphicallyinFig.2.6.Moreover,
1
(x);
3
(x)and
2

5
(x)are,continuousbutnotdierentiableatx=
1
.
2
Althoughelementsinthebasisofmultiwaveletfamilyovercomefewlimitations(e.g.,lossof
orthogonalityforelementswithpartialsupports)ofelementsinthebasiscomprisingtruncatedscale
functionsinDaubechiesfamily,basisinmultiwavelet

family

haveothershortcoming,viz.,fewscale
functions(wavelets)arediscontinuousattwoends
k
j;
k+1 k+
1
2
j(
j)oftheirsupportsasisevident
2 2 2
fromtheirgures.ItisthusdesirabletosearchforabasiscompatibletotheMRAofL
2
([a;b])
whoseelementsarefreefromdicultiesmentionedabove.
2.2.3Orthonormal(boundary)scalefunctionsandwavelets
ThekeyelementsofunconditionalbasisfortheMRAofL
2
(=[a;b]R)involvingscalefunctions
inDaubechiesfamilywithKvanishingmomentsoftheirwaveletsconsistsofsixsets(Cohenetal.,
1993;Anderssonetal.,1994;MonasseandPerrier,1998).Itisassumedthatsupp'is(;)=
(K+1;K).Therstthreesetsamongthesixinvolvethescalefunctions
left left left right right right
'
0
(x);'
1
(x);'
K1
(x);'(x);'
K
(x);'
K+1
(x);;'
1
(x) (2.2.3.1)
andtherestthreeinvolvethewavelets
left left left right right right

0
(x);
1
(x);;
K1
(x); (x);
K
(x);
K+1
(x); ;

(2.2.3.2)
1
(x):

37 2.2. MRA in L
2
(a, b)
The support of each of the
lef t
functions
l
x) and
lef t
ϕ( ψ
l
(x) (l = 0, 1, · · · , K − 1) is [0, 2K − 1] ⊂ R
+
while that of the functions ϕ
right
(x) and ψ
right
(x) (r = −K, −K + 1, · · · , −1) is [−2K + 1, 0] ⊂ R
−
r r .
The superscript left (right) is used to indicate that the support of the corresponding function contains
the left (right) end of the half line R
+
= [0, ∞) (R
−
= (−∞, 0]). All the above functions are deﬁned
at resolution 0. Their forms at resolution j ∈ Z are deﬁned as

lef t
j
l
j
( ) = 2
ef t
ϕ x ϕ (2
j
x); ϕ(x) = 2
ϕ(2
j
x −
right
x) 2
j
l);ϕ ( =
ϕ
right
2
l jl
2
jr
(2
j
2
jl r x), (2.2.3.3)
lef t
j
lef t j
j
right
j
ψ
2
right

j
2 2
j
jl
(x) = 2 ψ
l
(2x); ψjl(x) = 2 ψ(2x − l);ψ
jr
(x) = 2 ψ
r (2x). (2.2.3.4)
{
lef t lef t
The setϕ
j l
(x); l = 0, 1, · · · , K −1}∪{ϕj0k(x); k = K, K +1, · ·}∪[
0
· {ψ
jl
(x); l = 0, 1, · · · , K −1}∪
{ψjk(x); k = K, K+1, · · ·}, j ≥ j0] for some j0 ∈ Z forms an orthonormal basis for L
2
(R
+
). Similarly,
right
the set {ϕj0k(x); k = · · · , −K − 2, −K − 1} ∪ {ϕ
j r
(x); r = −K, −K + 1, · · · , −1} ∪ [{ψjk(x); k =
0
· · · , −K − 2, −K − 1} ∪ {
right
ψ
jr
(x); r = −K, −K + 1, · · · , −1}, j ≥ j0] for some j0 ∈ Z forms an
orthonormal basis for L
2
(R
−
). Contrary to the single relation (2.1.4.2) ϕ(x) = h · Φ(x) for the
reﬁnement equation for
lef t
ϕ(x), the two-scale relations for each of ϕ
l
(x), l = 0, · · · K − 1 and
ϕ
right
r (x), r = −K, · · · , −1 involve other functions of the corresponding set and some interior scale
functions adjacent to the respective boundary. We use the symbols (Panja and Mandal, 2015)
lef t lef t lef t lef t
Φ
j
(x) = (ϕ
j 0
(x), ϕ
j 1
(x), · · · , ϕ
j K1
(x))K,×1 (2.2.3.5a)
−
right right right right
Φ
j
(x) = (ϕ
j
(x), ϕ
−K j −K+1
(x), · · · , ϕ
j 1
(x)), (
− K 2.2.3.5b) ×1
lef t lef t lef t lef t
Ψ
j
(x) = (ψ
j 0
(x), ψ
j 1
(x), · · ·
Mandal
, ψ

NBand j K1
(x))K, (2.2.3.5c)×1
−
right right
PanjaM
right
M
right
Ψ
j
(x) = (ψ
j
(x), ψ ψ (x)), (2.2.3.5d)
−K j
(x), ,
−K+1
· · ·
j −1 K×1
Φ
LI
j (x) = (ϕj K (x), ϕj K+1(x), · · · , ϕj 3K2(x))
(2K1) 1, (2.2.3.5e) − − ×
Φ
RI
j (x) = (ϕj 3K+1(x), ϕj 3K+2(x), · · · , ϕj K1(x))
(2K1) 1. (2.2.3.5f)− − − − − ×
Then the two-scale relations for boundary scale functions can be stated as
√ �
lef t ef
Φ
0
= 2 H
lef tl t

(x) Φ
0
(2x) + H
LI
Φ
LI
0 (2x) , (2.2.3.6a)
�
right right
Φ
0
(x) =
√
2 H
right
Φ
0
(2x) + H
RI
Φ
RI
(2
�
0x) , (2.2.3.6b)
and the relation amongst boundary scale functions and wavelets can also b
�
e expressed as

lef t
Ψ(x) =
√
2 G
lef tlef t
0
Φ
0
(2x) + G
LI
Φ
LI
0 (2x) , (2.2.3.7a)

right right
Ψ
0
(x) =
√
2
�
�
G
right
Φ (2x) + G
RI
0
Φ
RI
(2
�
0 x)
�
. (2.2.3.7b)
In the above, the K × K matrices H
lef t
, H
right
, G
lef t
, G
right
and the K (2K 1) matri
ces H
LI
, H
RI
, G
LI
, G
RI
× −
are boundary ﬁlters (H’s are low-pass, G’s are high-pass with elements
lef t
h
right lef t right
km
, h
rm , g
k

m
, g
rm in the notation of Cohen et al., 1993. These ﬁlter coeﬃcients play an
important role in the MRA of L
2
([a, b]) and multiscale representation/regularization of singular
operators often appear in the subsequent chapters of the book. However, their determination is a

M
M Panja and B N Mandal

38 2.2. MRA in L
2
(a, b) φDau4
0
left
φDau4
1
left
φDau4
2
left
φDau4
3
left
1 2 3 4 5 6 7
-1
1
2
φDau4
k
left φDau4
0
right
φDau4
1
right
φDau4
2
right
φDau4
3
right
-6 -5 -4 -3 -2 -1 1
-1.0
-0.5
0.5
1.0
φDau4
k
right
Figure 2.7: Plots of orthonormal boundary scale functions for (a) at the left end of [0, ∞), (b) at
the right end of (−∞, 1] for ϕ in DauK4. ψDau4
0
left
ψDau4
1
left
ψDau4
2
left
ψDau4
3
left
1 2 3 4 5 6 7
-1.0
-0.5
0.5
1.0
1.5
2.0
ψDau4
k
left ψDau4
0
right
ψDau4
1
right
ψDau4
2
right
ψDau4
3
right
-6 -5 -4 -3 -2 -1 1
-1.0
-0.5
0.5
1.0
ψDau4
k
right
Figure 2.8: Plots of orthonormal boundary wavelets for (a) at the left end of [0, ∞), (b) at the right
end of (−∞, 1] for ϕ in DauK4.
separate issue. Theoretical aspects of determination of boundary ﬁlters were considered in detail
separately by Chui and Quak (Chui and Quak, 1992), Cohen et al. (Cohen et al., 1993), Andersson
et al. (Andersson et al., 1994), Monasse and Parrier (Monasse and Perrier, 1998). Their computa
tional aspects have been discussed by Chyzak et al. (Chyzak et al., 2001), Lee and Kassim (Lee and
Kassim, 2006) and Alt¨urk and Keinert (Alt¨urk and Keinert, 2012; Alt¨urk and Keinert, 2013).
Some representative of boundary scale functions and wavelets corresponding to the scale functions
in Daubechies family and symlets have been presented to observe their property in their respective
supports. From the careful analysis of Fig. 2.7 to Fig. 2.10 it appears that magnitude of values of
boundary scale functions and wavelets on the left end of the domain (0 in case of R
+
) are negligible
on a portion of their support both for Daubechies family and symlets. But, on the other end (0
in case of R
−
) boundary scale functions and wavelets corresponding to symlets maintain the same
behaviour (negligible in some part of their support), while the values the boundary elements (both
scale functions and wavelets) in Daubechies family are signiﬁcant there.
The regularity of the approximants may be estimated from the coeﬃcients in their representation
in the basis comprising of elements with compact supports involve in the MRA of L
2
([a, b]) with the
aid of the following theorem.
Theorem 2.2. (Cohen et al., 1993) For orthonormal basis having K vanishing moments of their
lef t
wavelets in MRA of L
2
([a, b]), choose j0 so that 2
j0
(b − a) ≥ 2K. Then the collection Φ
j0
(x −

� � ��
�
M
M Panja and B N Mandal

39 2.2. MRA in L
2
(a, b) φ
0
left
φ
1
left
φ
2
left
φ
3
left
1 2 3 4 5 6 7
-1.5
-1.0
-0.5
0.5
1.0
φSym
k
left φ
0
right
φ
1
right
φ
2
right
φ
3
right
-6 -5 -4 -3 -2 -1 1
-1.0
-0.5
0.5
1.0
φSym
k
right
Figure 2.9: Plots of orthonormal boundary scale functions for (a) at the left end of [0, ∞), (b) at
the right end of (−∞, 1] for ϕ in SymK4. ψ
0
left
ψ
1
left
ψ
2
left
ψ
3
left
1 2 3 4 5 6 7
-1.5
-1.0
-0.5
0.5
1.0
1.5
ψSym
k
left ψ
0
right
ψ
1
right
ψ
2
right
ψ
3
right
-6 -5 -4 -3 -2 -1 1
-1.5
-1.0
-0.5
0.5
1.0
ψSym
k
right
Figure 2.10: Plots of orthonormal boundary wavelets for (a) at the left end of [0, ∞), (b) at the
right end of (−∞, 1] for ϕ in SymK4.
∪
ht
a
rig
)ϕj0 k(x), k ∈ 2
j0
a + K, · · · , 2
j0
b

− K − 1 ∪ Φ
j
(b
0
−
l t
x) ∪
ef
j j0 Ψ
j
(x − a) ≥ ∪ {ψj k(x),
k ∈
�
2
j
a + K, · · · , 2
j
b − K − 1
��
∪
right
Ψ
j
(b − x) is an orthonormal basis for L
2
([a, b]). If r is
0
the Holder¨ index of interior scale function ϕ and wavelet
�
ψ (i.e., ϕ, ψ ∈ C
r
), then this collection is
an unconditional basis for C
s
([a, b]) for s < r.
We have therefore achieved our goal to have MRA of L
2
([a, b]) involving orthonormal basis
generated by
i) K boundary scale functions and wavelets each containing the left edge a,
ii) K boundary scale functions and wavelets each containing the right edge b,
iii) 2
j0
(b − a) − 2K interior scale functions,
iv) 2
j
(b − a) − 2K, j = j0, · · · , J − 1 interior wavelets,
resulting in a numerically stable procedure for the multiscale approximation of any function in
L
2
[a, b]. As on R, we have no explicit analytic expressions for the wavelets and scaling functions on
the interval [a, b]. For practical applications (getting multiscale representation or regularization of
singular operators) all that we really needed are the (boundary as well as interior) ﬁlter coeﬃcients.

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different content

In practicing riding in a circle, it will be found very advantageous to
vary the size of the circle, first riding in a large one, then gradually
contracting it, and again enlarging it; or the rider, while practicing
upon a large circle, may make a cross-cut toward the centre of this
circle, so as to enter upon another one of smaller diameter, and,
after riding for a short time in the smaller circle, she may again pass
out to resume her ride upon the larger one. These changes from
large to narrow circles form excellent practice for pupils, but should
always, if possible, be performed under competent instruction.
The first lessons in trotting in a circle should always be of short
duration, and the pupil required to ride slowly, the speed being
gradually increased as she gains knowledge and confidence. The
moment she experiences fatigue she should dismount, and rest,
before resuming the lesson.
In the amble the horse's movements very strongly resemble those
of the camel, two legs on one side moving together alternately with
the two legs of the other side. Thus one side of the animal supports
the weight of his body, while the other side moves forward, and so
on in alternation. This is an artificial gait, and one to which the horse
must usually be trained; though some horses whose ancestors have
been forced to travel in this gait, have themselves been known to
amble without any training. In the feudal ages it was the favorite
pace for a lady's palfrey, but at the present day it is no longer
countenanced by good taste.
The pace, however, which is so well liked by many ladies in this
country, is a kind of amble, although the steps taken are longer. A
good pacer can frequently travel faster than most horses can in the
trot. When the steed moves easily and willingly, the pace is very
pleasant for short rides, but for long journeys, unless the animal can

change his gait to a hand gallop or a canter, it will become very
unpleasant and tiresome. Many pacers are almost as rough in their
movements as the ordinary trotter; and although they do not jolt the
rider up and down upon the saddle, yet they jerk her body in such a
manner as successively and alternately to throw one side forward
and the other slightly back with each and every step, rendering a
ride for any distance very fatiguing.
The rack, at one time so much liked, has become almost obsolete.
This is a peculiar gait, not easily described, in which the horse
appears to trot with one pair of legs and amble with the other, the
gait being so mixed up between an amble and a defective trot as to
render it almost a nondescript. When racking, the horse will appear
constrained and uncomfortable, and will strongly bear upon the
rider's hand; some animals so much so, as completely to weary the
bridle-hand and arm in a ride of only an hour or two. This constant
bearing of the horse's head upon the reins soon renders him hard
mouthed, and, consequently, not easily and promptly managed. The
rack soon wears out a horse, besides spoiling him for other gaits,
and so injures his feet and legs that a racker will rarely be suitable
for the saddle after his eighth year. It is an acquired step, much
disliked by the horse, which has always to be forced into it by being
urged forward against the restraint of a curb-bit; and he will,
whenever an opportunity presents, break into a rough trot or canter,
so that the rider has to be constantly on the watch, and compel him
to keep in the rack against his will. And although the motion does
not jolt much, the aspect of the horse and rider is not as easy and
graceful as in the canter and hand gallop, there being an appearance
of unwillingness and restraint that is by no means pleasing. The
directions for the French trot will answer for both the pace and the

rack, except that in the latter the traction upon the reins must be
greater.

CHAPTER IX.
THE CANTER.
"When troubled in spirit, when weary of life,
When I faint 'neath its burdens, and shrink from its strife,
When its fruits, turned to ashes, are mocking my taste,
And its fairest scene seems but a desolate waste,
Then come ye not near me, my sad heart to cheer
With friendship's soft accents or sympathy's tear.
No pity I ask, and no counsel I need,
But bring me, oh, bring me my gallant young steed,
With his high arched neck, and his nostril spread wide,
His eye full of fire, and his step full of pride!
As I spring to his back, as I seize the strong rein,
The strength to my spirit returneth again!
The bonds are all broken that fettered my mind,
And my cares borne away on the wings of the wind;
My pride lifts its head, for a season bowed down,
And the queen in my nature now puts on her crown!"
Grace Greenwood .
In the gallop, the horse always has a leading foot or leg. In leading
with the right fore-foot, he will raise the left one from the ground,
and then the right will immediately follow, but will be advanced
somewhat beyond the left one; and this is the reason why, in this
case, the right side is called the "leading side." In the descent of the
fore-feet, the left one will touch the ground first, making the first
beat, and will be immediately followed by the leading or right fore-
foot which will make the second beat. The hind-legs are moved in a
similar way, the left hind-foot making the third beat, and the right
one the fourth. These beats vary in accordance with the adjustment

of the horse's weight, but when he gallops true and regular, as in the
canter, the hoof-beats distinctly mark one, two, three, four. In the
rapid gallop the hoof-beats sound in the time of one-two, or one-
two-three.
In leading with the left foot, the left side of the horse will be
advanced slightly and the left leg be carried somewhat beyond the
right, the action being just the reverse of that above described when
leading with the right leg. In this case the left side is termed the
"leading side." The hoof-beats of horses in the trot and gallop have
been admirably rendered by Bellini, in the opera of "Somnambula,"
just previous to the entrance of Rudolfo upon the stage. There are
three kinds of gallop, namely, the rapid or racing, the hand gallop,
and the canter.
The canter is a slow form of galloping, which the horse performs by
throwing his weight chiefly upon his hind-legs, the fore ones being
used more as supports than as propellers. Horses will be found to
vary in their modes of cantering, so much so as to render it almost
impossible to describe them accurately. Small horses and ponies
have a way of cantering with a loose rein, and without throwing
much weight upon their haunches, moving their feet rapidly, and
giving pattering hoof-beats. Most ponies on the Western prairies
canter in this manner, and it is said to be a very easy gait for a
horseman though very unpleasant, from its joltings, for a lady.
Another canter is what might be termed the "canter of a livery-stable
horse." This appears to be partly a run and partly a canter, a
peculiarity which is due to the fact that one or more of the animal's
feet are unsound, and he adopts this singular movement for the
purpose of obtaining relief. The little street gamins in London
recognize the sound of this canter at once, and will yell out, in time

with the horse's hoof-beats, "three pence, two pence," in sarcastic
derision of the lady's hired horse and the unhappy condition of his
feet.
In the true canter, which alone is suitable for a lady, the carriage of
the horse is grand and elegant. In this gait, the animal has his hind-
legs well under his body, all his limbs move regularly, his neck has a
graceful curve, and responds to the slightest touch of the rider's
hand upon the reins. A horse that moves in this manner is one for
display; his grand action will emphasize the grace of a finished rider,
and the appearance of the tout ensemble will be the extreme of
elegance and well-bred ease.
Horses intended for ladies' use are generally trained to lead in the
canter with the right or off fore-foot. Most lady riders, whose lessons
in riding have been limited, sit crosswise upon their saddles. This
position, without their being aware of it, places them more in unison
with the horse's movements, and thereby renders the canter with
this lead the easiest gait for them. But if a horse be constantly
required to canter with this lead he will soon become unsound in his
left hind-leg, because in leading with the right fore-foot he throws
the greater part of his weight upon his left hind-leg, and thus makes
it perform double duty. For this reason the majority of ladies' horses,
when the canter is their principal gait, will be found to suffer from
strained muscles, tendons, and articulations.
A finished rider will from time to time relieve her horse by changing
the lead to the left leg, or else she will change the canter to a trot.
Should her horse decidedly refuse to lead with the foot required,
whether right or left, it may be inferred that he is unsound in that
leg or foot; in which case he should be favored, and permitted to

make his own lead, while the canter should frequently be changed to
a walk.
Fig. 31.—Entering upon the Canter with the Right Leg leading.
To commence the canter, the horse must be brought to a walk, or
to a stand, then be placed on his haunches, and collected by means
of the curb, left leg, and whip; and then the bridle-hand must be
raised, while the second, third, and fourth fingers are moved to and
fro, so as to give gentle pulls upon the curb-reins, thus soliciting the
animal to raise his fore-feet. In performing these manœuvres, the
rider must be careful to direct the leg with which she desires her
horse to lead. This may be done as follows: If she desires to have
the right leg lead, the tension upon the left curb-rein must, just
before the animal rises to take his first step, be increased enough to
make him incline his head so far to the left that the rider can see his
left nostril, while, simultaneously, her left leg must press against his

side. By these means, the horse will be prompted to place himself
obliquely, with his head rather to the left, and his croup to the right.
The rider, if seated exactly in the centre of her saddle, must take a
position corresponding to that of the horse, by throwing her right hip
and shoulder somewhat forward, her face looking toward the
animal's head, while her body is held erect with the shoulders
gracefully inclined backward, and the hollow of the back well curved
inward. Any stiffness or rigidity of the body must be guarded against
in these movements and positions. The rider must hold herself in a
pliant manner, and yield to the motions of the horse. The left leg
must be held steady, the knee being placed directly underneath the
third pommel, and care must be taken not to press upon the stirrup,
as this will tend to raise the body from the saddle, and convey its
weight almost wholly to the left side.
The hands must be held somewhat elevated and steady, and, as the
horse advances, the tension on the reins must be even, so that the
fingers can feel every cadence of his step, and give and take with his
movements. Unlike the trot, in which the horse must be supported
by the snaffle, the canter will require the curb to sustain and keep
up his action. After the animal has started in the canter with the
right leg leading, should he incline too much to the left, the tension
upon the right rein must be increased, so as to turn his head more
to the right and bring him to the proper inclination for the lead of
the right leg. This correction must be effected gradually and lightly,
so as not to disturb the gait, or cause him to change his leading leg.
This canter with the right leg leading is very easy to learn, and will
not require much practice to master.
However, should the horse fail to obey these indications of the left
rein and leg, and start off in a false and disunited manner, as

explained under "the turn in the canter," another course should be
pursued, namely: the tension upon the right or off curb-rein must be
increased so as to bring the animal's nose to the right, as if he were
going to turn to the right on a curve, while at the same time the left
leg must be pressed against his side in order to have him carry his
croup slightly to the right. Now he must be made to lift his fore-feet
by increased tension on both curb reins, and then be urged forward.
As he advances, the hands should be extended a little to give him
more freedom in the spring forward, and he will then naturally lead
with the right side advanced. When once started in this gait, the
rider must equalize the tension upon the reins, having placed herself
in the saddle, in the manner explained for the canter. To have him
lead with the left leg, a similar but reversed course must be pursued,
using pressure with the whip, instead of the leg, to make him place
his croup to the left.
To canter with the left leg leading will be found more difficult to
acquire, and will demand more study and practice. The horse,
having been collected, must then be inclined obliquely to the right.
To accomplish this, the rider must increase the tension of the right
curb-rein, and press her whip against the animal's right side, which
will urge his head to the right and his croup to the left. In order that
the position of the rider's body may correspond with that of the
horse, her left hip and shoulder must be slightly advanced, in
precedence of her right hip and shoulder. It will be observed that the
manœuvring in this lead is similar to that in which the right leg
leads, except that the direction of the positions, of the management
of the reins, and of the horse's bearing during the canter is simply
reversed; in either lead, however, the tension or bearing upon the
reins, as the horse advances in the canter, must be equal.

It may be proper to state here that, as the amount of tension
needed upon the reins when cantering varies considerably with
different horses, some needing only the lightest touch, the rider will,
consequently, have to ascertain for herself how much will be suitable
for her horse. Some horses, after having fairly started in the canter,
will bend their necks so as to carry their chin closer to the throat,
while others again will extend the neck so as to carry the chin
forward. In the first instance, the reins will have to be shortened in
order to give the animal the proper support in the gait, as well as to
keep up the correspondence between his mouth and the bridle-
hand; in the latter they will require to be lengthened, to give him
more freedom in his movement. Should the reins be held too short,
or the rider's hand be heavy and unyielding, the horse will be
confined in his canter; should the reins be held too long, he will
canter carelessly, and will either move heavily upon his fore-legs, or
break into an irregular trot.
A rider may by attending to the following directions readily
determine whether her horse be leading with the leg she desires,
and also whether he be advancing in a true and united manner: If
he be moving regularly and easily, with a light play upon the reins in
harmony with the give and take movements of the hand, his head
being slightly inclined in a direction opposite to that of the leading
leg, and his action being smooth and pleasant to the rider, he will, as
a rule, be cantering correctly. But if he be moving roughly and
unevenly, giving the rider a sensation of jolting, if his head is inclined
toward the same side as that of the leading leg, and he does not
yield prompt obedience to the reins, then he is not cantering
properly, and should be immediately stopped, again collected, and
started anew. If necessary this course should be repeated until he
advances regularly and unitedly.

Some horses, after having fairly entered upon the canter, will change
the leading leg, and will even keep changing from one to the other,
at short intervals. This is a bad habit, and one that will never be
attempted by a well-trained animal, unless his rider does not
understand how to support him correctly and to keep him leading
with the required leg. A horse should never be allowed to change his
leading leg except at the will of his rider; and should he do so, he
should be chidden and stopped instantly, and then started anew.
If the rider when trotting rapidly wishes to change to a canter, she
must first moderate the trot to a walk, because the horse will
otherwise be apt to break from the trot into a rapid gallop. Should
he insist upon trotting, when it is desired that he should canter, he
must be stopped, collected with the curb-bit, as heretofore described
in the directions for commencing the canter, and started anew. This
course must be repeated every time he disobeys, and be continued
until he is made to canter.
It may be remarked here that, in the canter, whenever the horse
moves irregularly, advances heavily upon his fore-legs, thus
endeavoring to force his rider's hand, or when he fails to yield ready
obedience, he should always be stopped, collected, and started
anew,—repeating this course, if necessary, several times in
succession. Should the animal, however, persist in his disobedience,
pull upon the reins, and get his head down, his rider must, as he
moves on, gently yield the bridle-reins, and each time he pulls upon
them she must gradually, but firmly, increase the tension upon them,
by drawing them in toward her waist. This counter-traction must be
continued until the horse yields to the bridle and canters properly.
When he pulls upon the reins his rider in advancing her hands to

yield the reins should be careful to keep her body erect, and not
allow it to be pulled forward.
The turn in the canter. In turning to the right, if the horse is
leading with the inward leg, or the one toward the centre of the
circle of which the distance to be turned forms an arc, in the present
instance the right fore-leg which is followed by the right hind-leg, he
is said to be true and united, and will be able to make the turn
safely. Should the turn be made toward the left, the horse leading
with his inward or left fore-leg, followed by the left hind-leg, he will
likewise be true and united.
On the contrary, the animal will be disunited when, in cantering to
the right, he leads with the right fore-leg followed by the left hind-
leg, or when he leads with the left fore-leg followed by the right
hind-leg. In either case, from want of equilibrium in action and
motion, a very slight obstruction may make him fall.
In turning toward the left, in a canter, the horse will be disunited if
he leads with the left fore-leg followed by the right hind-leg, or if he
leads with the right fore-leg followed by the left hind-leg, as in the
preceding instance, he will be liable to fall. A horse is said to go false
when, in turning to the right, in the canter, he leads with both left
legs, or advances his left side beyond his right; also, when in
cantering to the left he leads with both right legs or advances his
right side beyond his left; in either of these false movements he will
be very liable to fall.
When it is desired to turn to the right, in the canter, the horse
must be kept well up to the bridle, so as to place his haunches
forward and well under him, thus keeping him light on his fore-legs,
and preventing his bearing too heavily upon his shoulders; and,

while the inward rein is being tightened in order to make the turn,
the outward one must continue to support the horse, being just
loose enough to allow him to incline his head and neck toward the
inner side of the turn. Pressure from the left leg of the rider will
keep the animal from inclining his haunches too much to the left,
during the turn. Should the steed be turned merely by means of the
inward rein, without being kept well up to the bridle, and without
either leg or whip being used upon his outer side, he will turn
heavily upon his forehand, and will be obliged to change to the
outward leg in order to support himself. This will cause him, after
the turn has been accomplished, to advance in a disunited way in
the canter.
When it is desired to turn to the left, the instructions in the
preceding paragraph may be pursued, the directions, however, being
reversed and pressure with the whip being employed instead of that
with the leg.
Sudden, sharp turns, are always dangerous, however sure-footed
the horse may be, and especial care should be taken not to turn
quickly to the right when the left fore-leg leads, nor to the left when
the right fore-leg leads, as in either case the animal will almost
certainly be thrown off his balance. In turning a "sharp corner,"
especially when the rider cannot see what she is liable to encounter,
it will be better for her to make the turn at a walk, and keep her own
side of the road, the right.
The stop in the canter. In bringing the horse to a stand, in the
canter, he should be well placed on his haunches by gradually
increasing the pull upon the curb-reins just as his fore-feet are
descending toward the ground; the hind-feet being then well under
the horse will complete the stop. The rider must guard against

leaning forward, as this will not only prevent the horse from
executing the stop in proper form, but should he suddenly come to a
stand, it will throw her still farther forward, and the reins will
become relaxed. Now, while she is thus leaning forward, should the
animal suddenly raise his head, the two heads will be very likely to
come into unpleasant contact; or should the horse stumble, his
liability to fall will be increased, because the rider will not be in a
proper position to support him, and will increase the weight upon his
shoulders, by being so far forward.
Many ladies not only lean forward while effecting the stop, but also
draw the bridle-hand to the left, and carry the bridle-arm back so
that the elbow projects behind and beyond the body, while at the
same time they elevate the shoulder on this side. This is an
extremely awkward manner of bringing a horse to a stand. The stop
should be made in the same manner as that described in the walk,
that is, by gradually drawing the bridle-hand toward the waist, etc.
Nearly all horses, unless exceptionally well trained, will trot a short
distance before coming to a stand in the canter or gallop, and it is
here that a knowledge of the French or cavalry trot will prove
essential, because the rider will then comprehend the motion, and
will sit closely to the saddle until the horse stops. In all cases, the
horse should be brought to a stand in a regular, collected manner, so
that with a little more liberty of rein he can promptly reënter upon
the canter, should this be desired.

CHAPTER X.
THE HAND GALLOP.—THE FLYING GALLOP.
"Now we're off like the winds to the plains whence they came;
And the rapture of motion is thrilling my frame!
On, on speeds my courser, scarce printing the sod,
Scarce crushing a daisy to mark where he trod!
On, on like a deer, when the hound's early bay
Awakes the wild echoes, away, and away!
Still faster, still farther, he leaps at my cheer,
Till the rush of the startled air whirs in my ear!
Now 'long a clear rivulet lieth his track,—
See his glancing hoofs tossing the white pebbles back!
Now a glen dark as midnight—what matter?—we'll down
Though shadows are round us, and rocks o'er us frown;
The thick branches shake as we're hurrying through,
And deck us with spangles of silvery dew!"
Grace Greenwood .
The hand gallop is an intermediate gait between the canter and the
flying gallop. Its motion, though rather rapid, is smooth, easy, and
very agreeable for both rider and steed. Nearly all horses, especially
spirited ones, prefer this movement to any other; the bronchos on
the plains of the far West will keep up this long, easy lope or hand
gallop for miles, without changing their gait, or requiring their riders
to draw rein, and without any apparent fatigue. This pace is likewise
a favorite one with riding parties, as the motion is so smooth that
conversation can be kept up without difficulty. If the animal's
movements are light, supple, and elegant, the lady rider presents a
very graceful appearance when riding this gait, as the reactions in it
are very mild; it is the gait par excellence, for a country ride.

On a breezy summer morning, there is nothing more exhilarating
than a ride at a hand gallop, on a willing, spirited horse; it brightens
the spirits, braces the nerves, refreshes the brain, and enables one
to realize that "life is worth living."
"I tell thee, O stranger, that unto me
The plunge of a fiery steed
Is a noble thought,—to the brave and free
It is music, and breath, and majesty,—
'Tis the life of a noble deed;
And the heart and the mind are in spirit allied
In the charm of a morning's glorious ride."
Let all gloomy, dyspeptic invalids try the cheering effects of a hand
gallop, that they may catch a glimpse of the sunlight that is always
behind even the darkest cloud of despondency.
When the horse is advancing in a collected canter, if the rider will
animate him a little more by gentle taps with the whip, and then as
he springs forward give him more liberty of the curb-rein, he will
enter upon a hand gallop. In this gait he will lead either with the
left or the right foot, but the oblique position of his body will be very
slight. The management of the reins, the turns to the right or to the
left, the stop, and the position of the rider's body, must, in this gait
be the same as in the canter, except that the body need not be quite
so erect, and the touch upon the reins must be very light, barely
appreciable.
If riding a spirited horse, the lady must be upon her guard, lest he
increase his speed and enter into a flying or racing gallop. Any horse
is liable to do this when he has not been properly exercised,
especially if he is with other horses, when a spirit of rivalry is
aroused, and he sometimes becomes almost unmanageable from
excitement. Many livery-stable horses, although quiet enough in the

city, will, when ridden upon country roads, especially in the spring,
require all the skill of their riders to keep them under control. The
change from the stone and brick of the city or town to the odor of
the fresh grass and the sight of green fields has an exhilarating
effect upon them, and makes them almost delirious with gladness,
so that they act like anything but sensible, quiet, well-worked
horses.
When her horse manifests any such disposition, the rider must retain
her presence of mind, and not permit any nervousness or
excitement on her part to increase that of her horse. She must keep
him well under the control of the curb-bit, and not allow him to
increase his speed; when he endeavors to do so, she must sit erect,
and every time his fore-feet touch the ground she must tighten the
curb-reins, by drawing them gradually but firmly toward her waist.
She will thus check the animal's desire to increase his speed, by
compelling him to rest upon her hand at short intervals until he can
be brought under command and again made obedient. Care must be
taken not to make this strong pull upon the animal's mouth
constant, as this will be more apt to increase than to lessen his
speed, and will also prevent her from turning him readily should she
encounter any object upon the road.
Should the horse, however, continue to disobey the commands of his
rider, and persist in his efforts to increase his speed, she must then
lean well back, and "saw his mouth" with the snaffle-reins, that is,
she must pull first one of these reins and then the other in rapid
succession; this may cause him to swerve out of a straight course,
but if he has a snaffle-bit separate from the curb this sawing will
generally have the desired effect, and stop him.

If the horse should get his head down and manifest a disposition to
change the full gallop into a runaway, the rider must, as she values
her own safety, keep her body well inclined backward, for some
horses, when excited, will, while their riders are endeavoring to
check or control them, kick up as they gallop along, and the rider,
unless she is prepared for such movements, will be in danger of
being thrown. In such a case every effort must be made to raise the
horse's head. To do this, the rider must slacken the curb-reins for a
moment, and then suddenly give them a strong, decided jerk
upward; this will cause a severe shock to the horse's mouth, and
make him raise his head and stop suddenly, a movement that may
throw her toward or upon the front of the saddle with considerable
force, unless she guard herself against such an accident by leaning
well back.
Should the horse, when galloping at full speed, turn a corner in spite
of the efforts of his rider, she must keep a steady pull upon the outer
curb-rein, and lean well back and in toward the centre of the curve
which the horse is describing in his turn. All this must be done
quickly, or she will lose her balance and fall off upon the outer side.
During all these violent efforts of the horse the rider must keep a
firm, steady seat, pressing her left knee up strongly against the third
pommel, and at the same time holding the second clasped firmly by
the bend of her right knee. If she recollects to do all this, there will
be little cause for alarm, as it will then be very difficult for her horse
to unseat her. The combined balance and grip of limbs will give her a
firmer seat than it is possible for a man to acquire in his saddle.

Fig. 32.—The Flying Gallop.
In the flying or racing gallop the horse manifests the utmost
capabilities of his speed, his body at every push of his hind-legs
being raised from the ground so quickly that he will appear as if
almost flying through the air; hence the name "flying gallop." In this
gait it is unimportant with which leg the horse leads, provided the
advance of the hind-leg on the same side as that of the leading one
be made correspondingly. It is advisable that every lady rider should
learn to sit the flying gallop, as she will then be better able to
maintain her seat, and to manage her horse should she ever have
the misfortune to be run away with. (Fig. 32.)
Many ladies, when riding in the country, enjoy a short exhilarating
flying gallop; and for their benefit a few instructions are here given
that will enable them to indulge their penchant for rapid riding,
without danger to themselves, or injury to their horses. Before the
lady attempts rapid riding, however, she must be thoroughly trained
in all the other gaits of the animal, must possess strong, healthy

nerves, and must have sufficient muscular power in her arms to hold
and manage her horse, and to stop him whenever occasion requires;
she must also have fitted to his mouth a curb-bit which possesses
sufficient power to control him and to bring him to a stand, when
this is desired. Above all, her horse must be sure-footed, and free
from any and every defect that might occasion stumbling.
Every point having been carefully attended to, and the lady being
ready for the ride, she must sit firmly upon the centre of the saddle,
grasping the second and third pommels, as described above. She
must be careful not to press strongly upon the stirrup, as this will
tend to raise her body from the saddle. From the hips down the
body and limbs must be held as immovable as possible. The body,
below the waist, must by its own weight, aided by the clasp of the
right and left legs upon their respective pommels, secure a firm seat
upon the saddle. From the waist up the body must be pliable, the
shoulders being well back, and the back curved in, so that the rider
may keep her balance, and control the horse's action. The reins
must be held separately, in the manner described for holding the
double bridle-reins in both hands. The animal must be ridden and
supported by the snaffle-reins, the curb being held ready to check
him instantly should he endeavor to obtain the mastery. The hands
must be held low, and about six or eight inches apart, and the rider's
body must lean back somewhat.
Leaning forward is a favorite trick of the horse-jockey when riding a
race, as it is supposed to assist the horse, and also enable the rider
to raise himself on the stirrups; but as lady riders are not horse-
jockeys, and are not supposed to ride for a wager, but simply for the
enjoyment of an exhilarating exercise, it will not be at all necessary
for them to assume this stooping posture. Many of the best

horsemen, when riding at full gallop in the hunting field, or on the
road, prefer to incline the body somewhat backward, this having
been found the safest as well as most graceful position for the rider.
As the horse moves rapidly forward, the rider, while keeping a firm
hand upon the snaffle-reins so as to give full support to the horse,
must be sure with every stride of the animal to "give and take," and
this motion, instead of being limited to the hands and wrists, as in
all other gaits, must in this one embrace the whole of the fore-arms,
which, using the elbows as a hinge, should move as far as is
necessary.
To stop the horse in a flying gallop, the curb-reins must be drawn
upward and toward the waist gradually, for should they be pulled
upon suddenly it would be apt to stop him so abruptly that he would
either become overbalanced, or cross his legs, and fall.
In this gait, the rider should never attempt to turn her horse except
upon a very large circle, because, even when in the proper position,
unless she possesses great muscular power, she will be almost
certain to be thrown off on the outward side by the forcible and
vigorous impetus imparted.

CHAPTER XI.
THE LEAP.—THE STANDING LEAP.—THE FLYING
LEAP.
"Soft thy skin as silken skein,
Soft as woman's hair thy mane,
Tender are thine eyes and true;
All thy hoofs like ivory shine,
Polished bright; oh, life of mine,
Leap, and rescue Kurroglou!"
Kyrat, then, the strong and fleet,
Drew together his four white feet,
Paused a moment on the verge,
Measured with his eye the space,
And into the air's embrace
Leaped as leaps the ocean serge.
Longfellow,
The Leap of Roushan Beg.
A lady rider who has the nerve and confidence to ride a hand gallop,
or a flying gallop, will be ready to learn to leap. Indeed, instruction
in this accomplishment should always be given, as it is of great
assistance in many emergencies. The most gentle horse may
become frightened, shy suddenly to one side, or plunge violently for
some reason or other, and these abrupt movements strongly
resemble those of leaping; if, therefore, the rider understands the
leap, she will know better how to maintain her equilibrium. Or she
may meet some obstruction on the road, as the trunk of a tree felled
by a storm; when, instead of being compelled to return home
without finishing her ride, she can leap over the obstacle. Again,

should she at any time be in great haste to reach her destination she
may, by leaping some low gap in a fence, or some small stream, be
able to take one or more short cuts, and thus greatly lessen the
distance she would have had to ride on the road.
Leaping is by no means difficult to learn. With an English saddle, the
third pommel will prevent the rider from being shaken off by the
violence of the motion, and will thus make leaping entirely safe for a
lady provided the horse be well-trained and sure-footed. Before
venturing upon a leap, three requisites are necessary: first, the
horse must be a good and fearless leaper; second, the rider must
have confidence in herself and steed, because any nervousness on
her part will be apt to cause the animal to leap awkwardly; and
third, she must always be sure of the condition of the ground on the
opposite side of the object over which the leap is to be made—it
must neither slope abruptly down, nor present any thorny bushes,
nor be so soft and soggy that the horse will be apt to sink into it. No
risk must be taken in the leap, except in cases of emergency, when,
of course, the rider may have neither time nor opportunity to select
her ground, and be obliged to leap her steed over the nearest
available point. The author once avoided what might have proved a
serious accident to both herself and horse, by promptly leaping him
over a hedge of thorn bushes, upon the other side of which was a
river: this was done in order to avoid colliding in a narrow road with
a frightened, runaway team, which was quite beyond the control of
its driver.

Fig. 33.—The Standing Leap—Rising.
The standing leap will prove more difficult to learn than the flying
leap, but, nevertheless, it should be the first one practiced, and
when once acquired, the other will be mere play. A bar twelve feet
long, raised two feet from the ground, will be sufficient for practice
in this exercise; if a lady can manage a leap of this height with
expertness and grace, she will be fully able to bound over a still
higher obstacle, should she desire to do so, and her horse be equal
to the occasion. Before attempting the leap, she must be sure that
she is perfectly secure upon the saddle, with her left knee directly
under the third pommel so as to press it firmly against the latter as
the horse rises to the leap; her left leg, from the knee to the stirrup,
must hang perpendicularly
8
along the side of the horse, the inner

surface or side of the knee lightly pressing against the saddle-flap;
her foot must be well placed in the stirrup; her seat directly in the
centre of the saddle; her body erect and square to the front; her
shoulders well back; and the small of her back curved in. The right
leg must firmly grasp the second pommel as the horse rises, and the
right heel be held somewhat back, and close to the fore-flap of the
saddle. The hands must be held low, and about six inches apart,
with a snaffle-rein in each, and the curb-reins must be so placed
that the rider will not unconsciously draw upon them, but must not
hang so loosely as to become caught accidentally upon any
projecting article with which they may come in contact. If all these
points be carefully attended to, just previous to walking the horse up
to the bar, the rider will be in correct position and ready for the leap,
which she will accomplish very quickly, with perfect security, and
with a much firmer seat than that obtained by the most finished
horseman.
The principal movement for which the rider should be prepared in
leaping is that of being thrown forward on the saddle, both when
the horse makes the spring and when his fore-feet touch the
ground. In order to avoid this accident, the rider, keeping a firm seat
and grasp upon the pommels, must incline her shoulders somewhat
backward, both when the horse springs from the ground and also
during the descent, the amount of inclination varying with the height
of the leap. The erect position should be resumed when the hind-
legs have again touched the ground. In a very high leap, the rider's
body should be bent so far back during the descent as to look
almost as if in contact with the back of the horse.
When the points named above have been attended to, the horse
must be collected, with his hind-legs well under him, and then be

briskly walked up to the bar or obstacle to be leaped and placed
directly before it, but not so close that he cannot clear it without
striking his knees against it as he rises,—sufficient room must always
be allowed him for his spring. Now, after receiving a light touch or
pull upon the reins to tell him that his rider is ready, he will raise
himself upon his hind-legs for the leap. As he rises, the rider's body,
if properly seated, as heretofore explained, will naturally assume a
sufficient inclination forward without any effort on her part. While in
this position she must not carry her shoulders forward, but must
keep them well back, with the waist well curved in as when sitting
erect. It should never be forgotten that in the rise during the leap,
just previous to the spring, no efforts whatever must be made by
the rider to support the horse, or to lift him, but instead, she should
simply hold the reins so lightly that his mouth can just be felt, which
is called "giving a free rein." If the reins be allowed to hang too
loosely they may catch upon some object not noticed by the rider,
and not only be wrenched from her hands, but also give the horse's
mouth a severe jerk, or perhaps throw him upon the ground. Too
loose a rein would, moreover, be apt to make it impossible for her to
give timely support to the animal as his fore-feet touched the
ground. The leap, it must be borne in mind, is effected very quickly.
(Fig. 33.)
As the horse springs from his hind-legs to make the leap, the rider
must advance her arms, with her hands held as low as possible so
as to give him a sufficiently free rein to enable him to extend
himself; this position of the arms will also prevent the reins from
being forcibly wrested from her hands by the horse's movements. At
the moment of the spring and the advance of the arms, the rider's
body must be inclined backward, the erect position of the waist and
shoulders being, however, maintained. As the animal's fore-feet

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Wavelet Based Approximation Schemes For Singular Integral Equations 1st Edition Madan Mohan Panja

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Wavelet Based Approximation Schemes For Singular Integral Equations 1st Edition Madan Mohan Panja

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