The Wavelet Transform (Atlantis Studies in Mathematics for Engineering and Science)

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ISBN 978-90-78677-26-0

I

9 789078 677260

ATLANTIS STUDIES IN MATHEMATICS FOR ENGINEERING AND SCIENCE VOLUME 4 SERIES EDITOR: C.K. CHUI

Atlantis Studies in Mathematics for Engineering and Science Series Editor: C. K. Chui, Stanford University, USA

(ISSN: 1875-7642)

Aims and scope of the series The series 'Atlantis Studies in Mathematics for Engineering and Science (AMES) publishes high quality monographs in applied mathematics, computational mathematics, and statistics that have the potential to make a significant impact on the advancement of engineering and science on the one hand, and economics and commerce on the other. We welcome submission of book proposals and manuscripts from mathematical scientists worldwide who share our vision of mathematics as the engine of progress in the disciplines mentioned above. All books in this series are co-published with World Scientific. For more information on this series and our other book series, please visit our website at: www.atlantis-press.com/publications/books

. , ATLANTIS ., PRESS AMSTERDAM -PARIS

w-

World Scientific

@ ATLANTIS PRESS I WORLD SCIENTIFIC

The Wavelet Transform

Ram Shankar PATHAK Department of Mathematics Banaras Hindu University Varanasi, India

~ATLANTIS

.,

PRESS

AMSTERDAM - PARIS

11»

World Scientific

Atlantis Press

29, avenue Laumiere 75019 Paris, France For information on all Atlantis Press publications, visit our website at: www.atlantis-press.com Copyright

This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher.

Atlantis Studies in Mathematics for Engineering and Science

Volume I: Continued Fractions: Volume I: Convergence Theory- L. Lorentzen, H. Waadeland Volume 2: Mean Field Theories and Dual Variation - T. Suzuki Volume 3: The Hybrid Grand Unified Theory- V. Lakshmikantham, E. Escultura, S. Leela

ISBN: 978-90-78677-26-0 ISSN: 1875-7642

@ 2009 ATLANTIS PRESS I WORLD SCIENTIFIC

To my wife Indrawati Pathak

Preface

The wavelet transform has emerged as one of the most promising function transforms with great potential in applications during the last four decades. The present monograph is an outcome of the recent researches by the author and his co-workers, most of which are not available in a book form. Nevertheless, it also contains the results of many other celebrated workers of the field. The aim of the book is to enrich the theory of the wavelet transform and to provide new directions for further research in theory and applications of the wavelet transform. The book does not contain any sophisticated Mathematics. It is intended for graduate students of Mathematics, Physics and Engineering sciences, as well as interested researchers from other fields. The Fourier transform has wide applications in Pure and Applied Mathematics, Physics and Engineering sciences; but sometimes one has to make compromise with the results obtained by the Fourier transform with the physical intuitions. The reason is that the Fourier transform does not reflect the evolution over time of the (physical) spectrum and thus it contains no local information. The continuous wavelet transform (W111 f)(b,a), involving wavelet lfl, translation parameter b and dilation parameter a, overcomes these drawbacks of the Fourier transform by representing signals (time dependent functions) in the phase space (time/frequency) plane with a local frequency resolution. The Fourier transform is restricted to the domain LP (JRn) with I defined for I

~

~ p ~

2, whereas the wavelet transform can be

p < oo with appropriate kernel lfl. The frequency resolution is controlled by

dilation parameter a, and for small

Ia I, (W111 j)(b,a) represents high frequency components

of the signal f. Hence, it is desirable to know asymptotic expansions of (W111 f)(b,a) for small and large values of the parameters. The convolution operation associated with an integral transform increases considerably the applicability and theory of the transform. Motivated from the work of 1.1. Hirschman vii

viii


Jr. on Hankel convolution, a theory of wavelet convolution is developed. This provides the interesting property that the product of two wavelet transforms could be a wavelet transform. The book assumes that the reader has a background in the elements of analysis. Chapter I essentially deals with the perquisite material for the theory of distributions and certain integral transforms and related topics. It gives a brief idea about wavelets and wavelet transforms. In Chapter 2 using properties of Fourier transform certain approximation properties of the wavelet transform are obtained. Certain relations between wavelet transform, Hilbert transform, generalized Hilbert transform and Riesz fractional integrals, and also between their generalizations are obtained. These relations are used to derive new inversion formulae for the wavelet transform of functions belonging to LP -space under different conditions. Inversion formulae, Parseval formulae, boundedness and approximation results are also obtained in certain weighted LP -spaces. Some of the results are extended to distributions. In Chapter 3 it is shown that the wavelet transform is a continuous linear map of the Schwartz space Y' (ffi.n) into a similar space Y' (ffi.n x ffi.+) when the wavelet belongs to

Y'(ffi.n). The composition of two wavelet transforms is defined and the continuity of the composition operator is investigated. A reconstruction formula for the composition of the wavelet transforms is obtained. Distributional extensions of these transforms are given. Boundedness results regarding these transforms on certain weighted Sobolev spaces are obtained. Abelian theorems are of considerable importance in solving boundary value problems of Mathematical Physics. Abelian theorems for various integral transformations are available in the literature. In this chapter we establish Abelian theorems for the wavelet transform of functions and afterwards derive certain distributional results. Chapter 4 is devoted to the study of the continuous wavelet transform on certain Gel'fandShilov spaces of typeS. It is shown that, for wavelets belonging to the one type of S-space defined on ffi., the wavelet transform is a continuous linear map of the other type of the Sspace into a space of the same type (latter type) defined on ffi. x ffi.+. The wavelet transforms of certain ultradifferentiable functions are also investigated. Chapter 5 contains a study of the continuous wavelet transform on certain Gel'fand-Shilov spaces of type W. The continuity and boundedness results for continuous wavelet transform are obtained on some suitably designed spaces of type W defined on ffi. x ffi.+, C x ffi.+ and

CxC.

Preface

ix

In Chapter 6 the continuous wavelet transform is studied on the generalized Sobolev space B~k·

Roundedness results in this Sobolev space are obtained. Local generalized Sobolev

space is defined and some of its important properties are discussed. Wavelet transform with compactly supported wavelet is also studied. Generalized translation and convolution operators for a general integral transform are defined in order to develop a unified theory of convolutions for all commonly used integral transforms. Thereby, definitions of translation and convolution for continuous wavelet transforms are given and their properties are investigated. A brief account of translation and convolution for discrete wavelet transform is also presented. A basic function D(x,y, z) associated with general wavelet transform is defined and its properties are investigated. Using D(x,y, z), translation and convolution associated with the wavelet transform are defined and certain existence theorems are proved. An approximation theorem involving wavelet convolution is also proved in Chapter 7. Chapter 8 contains derivation of a relation between the convolution associated with the wavelet transform and convolution associated with the Fourier transform. This relation is used to define the wavelet convolution transform and study its properties. Existence theorems are proved. We show that the product of two wavelet transforms could be a wavelet transform. Application of the wavelet convolution in approximation of functions is given. In Chapter 9 asymptotic expansion of the wavelet transform of a function f with respect to the wavelet II' is derived when the dilation parameter belongs to an open subset of (O.oo) and the translation parameter goes to infinity. It is assumed that both

f and II' possess

power series representations in descending powers oft. Asymptotic expansion of Mexican hat wavelet transform is obtained. Following Wong's technique asymptotic expansion of the wavelet transform is derived when translation parameter is fixed and dilation parameter goes to infinity and also to zero. Asymptotic expansions for Morlet wavelet transform, Mexican hat wavelet transform and Haar wavelet transform are obtained as special cases in Chapter 10. The author received inspiration and guidance from Professor C. K. Chui in writing this monograph. His sincere thanks are due to Professor Chui, the Editor of AMES series and Dr. Z. Karssen, Publisher Atlantis Press. The author is also thankful to Professor K. K. Azad (University of Allahabad) for his assistance in so many ways. The author's coworkers G. Pandey, A. Pathak and S. Verma were of great help in the preparation of the

X


monograph. In preparation of the manuscript the author received assistance from Mr. P. K. Sinha, Ms. Shikha Gaur and staff of Pushpa Publishing House, Allahabad (India). The research work was supported by the U.G.C.-Emeritus Fellowship (New Delhi). R. S. Pathak

Contents

vii

Preface 1.

1.1 1.2 1.3 1.4 1.5 1.6 1.7

1.8

2.

3.

1

An Overview Introduction . Introduction to distribution theory The Fourier transform . . . . . . . The Fourier transformation in Y'(!Rn) 1.4.1 Operation-transform formulae . The Hilbert transform . . . . . . . Wavelets . . . . . . . . . . . . . . . . . The continuous wavelet transform . . . 1.7.1 Relationship with Fourier transform 1.7.2 Parseval relation for the wavelet transform 1.7.3 Reconstruction formula . . . . 1.7.4 The discrete wavelet transform . Asymptotic expansion . . . . . . . . . . 1.8.1 The Mellin transform technique . The distributional approach 1.8.2

I 5 6 7 8 9 12 14 14 15 15 16 18

19

The Wavelet Transform on lJ'

21

2.1 2.2 2.3

21 26

2.4 2.5

Introduction . . . . . . . Approximation properties . . . . . . . . . . . . . . . . . . . . Wavelet transform, Hilbert transform and Fractional integrals . 2.3.1 Inversion formulae . . . . . . . . . . . . . . . . . . . 2.3.2 Approximation . . . . . . . . . . . . . . . . . . . . . 2.3.3 lnvariance under rational transformations of translation parameter . Wavelet transform on weighted LP- spaces . . . . . . . . . . . . . . Boundedness of the wavelet transform on LP with different weights

2.6 2.7 2.8

Wavelet transform and operators The Wavelet transform on F~ . . . The wavelet transform on lJ' (!Rn)

41 44 47

H(v), Htk and Itk

Composition of Wavelet Transforms 3.1 3.2 3.3 3.4

29 32 34 35 37 39

49

Introduction . . . . . . . . . . . The wavelet transform of tempered distributions . Composition of wavelet transforms . Weighted Sobolev Spaces . . . . . . xi

49 50 52 56


xii

3.5 3.6 3.7 3.8

4.

5.

6.

. . . .

59 61 65 67

4.1

Introduction . . . . . . . . . . . . . . .

67

4.2 4.3 4.4 4.5 4.6

Wavelet Transform on Spaces of Type S The Wavelet Transform of Generalized Functions The Wavelet Transform of Tempered Ultradistributions Wavelet Transform of Gevrey Functions of Compact Support . Band Limited Wavelets . . . . . . . . . . . . . . . . . . . . .

69 74 75

78 80

The Wavelet Transform on Spaces of Type W

83

5.1 5.2 5.3 5.4

83 83

Introduction . . . . . . . . . . . . . . . . The spaces WM,a, wn.,B and w:i:~ ... . The wavelet transformation on W -spaces . Examples . . . . . . . . . . . . . . . . .

86 91

93

The Wavelet Transform on a Generalized Sobolev Space

6.3

6.4 6.5 6.6

Introduction . . . . . . . . . . . . . The generalized Sobolev space Bmk . . . . . . . . . p, 6.2.1 Examples . . . . . . . . . . . . . . . . . . The wavelet transform on generalized Sobolev space B~k . 6.3.1 Adjoint of L'l' . . . . . . . . . . . . . . . 6.3.2 Bm k -approximation of wavelet transforms Asymptotfc' behaviour for small dilation parameters Local convergence Example . . . . . .

93

94 95 96 97 99 100 103 106

A Class of Convolutions: Convolution for the Wavelet Transform

109

7.1 7.2 7.3 7.4 7.5 7.6 7.7

109 109 112 118 120 122 127

Introduction . . . . . . . . . . . . . . . . . The generalized translation and convolution Special cases . . . . . . . . . . . . . . . . Convolution for the wavelet transform .. . Convolution for the discrete wavelet transform . Existence theorems . . . . . An Approximation Theorem

129

8. The Wavelet Convolution Product 8.1 8.2 8.3 8.4 8.5

9.

57

The Wavelet Transform on Spaces of Type S

6.1 6.2

7.

Wavelet transforms on weighted Sobolev spaces . . . . Abelian theorems for wavelet transform of functions . Abelian theorems for wavelet transform of distributions An Application . . . . . . . . . . . . . . . . . . . . .

Introduction . . . . . . . . . The Wavelet Convolution Product Existence Theorems . . . . . . . . Wavelet Convolution in a Generalized Sobolev Space Approximations in U' and B p,k Spaces . . . . . . . .

lbl is Large

137

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotic Expansions of the Generalized Stieltjes Transform .

137 142

Asymptotic Expansions of the Wavelet Transform when 9.1 9.2

129 130 132 135 135

Contents

9.3 9.4 9.5

10.

xiii

Asymptotic expansions for Tz and T3 . . . . . . . . . . . . Asymptotic Expansions of f*Syg and W(x,y) . . . . . . . Asymptotic Expansion of Mexican Hat Wavelet Transform

Asymptotic Expansions of the Wavelet Transform for Large and Small Values of a 10.1 10.2 10.3 10.4 10.5 I 0.6 10.7

Introduction . . . . . . . . . . . . . . . . . . . . . Asymptotic expansion for large a . . . . . . . . . . Asymptotic expansion of Morlet wavelet transform Asymptotic Expansion of Mexican Hat Wavelet Transform Asymptotic expansion of Haar wavelet transform Asymptotic expansion for small a . . . . . Asymptotic expansion for small a continued 10.7.1 The case It= 1 . . . . . . . . . .

143 145 150

155

155 156 158 161 162 164 168 169

Bibliography

171

Subject Index

175

Chapter 1

An Overview

1.1

Introduction

In this chapter we present some elements of theory of classical and distributional Fourier and Hilbert transforms that are to be exploited in the development of the theory of the wavelet transform in U -spaces ( 1 ::::;; p < oo) and in certain distribution spaces. For this purpose an elementary theory of distributions is given that is essential for proper understanding of various developments in the theory of wavelet transform presented in this book. The relationship between the wavelet transform and Fourier transform is well known. We shall find relations of the wavelet transform with Hilbert transform and Riesz fractional integral operator. The application of Fourier transform restricts itself to LP-spaces with I ::::;; p::::;; 2 but Hilbert transform approach gives results valid in U-spaces with (1::::;; p < oo). Hilbert transform has applications in signal processing, aerofoil problems, dispersion relations, high-energy physics, potential theory problems and others [53]. Therefore, using aforesaid relation the wavelet transform can also be applied to tackle all such problems. A brief idea about convolution operator, and asymptotic expansion of a general integral transform is also given, that will form a basis for development of corresponding results for the wavelet transform.

1.2

Introduction to distribution theory

In this section we give a brief account of the theory of distributions (or generalized functions) that we shall need in subsequent chapters. For more thorough discussion of these ideas we may refer to Friedman [20], Zemanian [97, 98].

Definition 1.2.1. Let K be a compact subset of JR.n. Let ~K be the set of all complex-valued infinitely differentiable functions on JR.n that vanish at all points outside K.

~K

is a linear

2


space under addition and multiplication by a complex number; the zero element in

~K

is

the identically zero function. The union of all by

~-

~K,

The space

where K varies through all possible compact subsets of IR.n, is denoted

~

is also a linear space under the usual definitions of addition and

multiplication by a complex number. Identically zero function is the zero element of also.

~

~

is called a test function space.

Example 1.2.2. An element in

~(IR.n)

is given by

cfJ(x) = { exp( lxiLI ), lxl < I 0, lxl ~ l. For each non-negative integer k E N 0 define a seminorm Yk on the linear space

(1.2.1) ~K,

as

follows:

Yk(C/J) :=sup ID*cp(x)l < oo, cfJ

E ~K,

(1.2.2)

xE~"

where D* = (a I axl )k 1 (a I ax2)k 2 ... (a I axn )kn. The topology of ~K is generated by the sequence of semi norms { Yk}, k E N0. The space

~K

is a complete countably multinormed

space. Definition 1.2.3. The space of all continuous linear functionals on ~K is denoted by ~~.

also called dual of ~K- ~~ is the dual of the space~- Its elements are called Schwartz distributions. The space of all those distributions in ~' which have compact support is denoted by lff'. Example 1.2.4. Dirac delta function D concentrated at xo, defined by

(D(x-xo),cp(x)) := cp(xo), cfJ E

~(l~n),

( 1.2.3)

is an example of a distribution. Example 1.2.5. Let K be a compact subset of IR.n, and let f(x) be a locally integrable func-

tion on IR.n. Define the functional f on

~K

(f,cp) :=

by

r f(x)cp(x)dx,

}~n

(1.2.4)

where cfJ E ~K.Then f E ~~Differentiation of a distribution is a basic property which has significant applications. Definition 1.2.6. Distributional derivative~ ox off E ~'(IR.n) is defined by 1

I at ) I aq,) . n \ax;'q, =-\f'axj ,J=l,2, ... ,n, C/JE~(IR. ).

(1.2.5)

An Overview

3

· I ""(JX:• of J·- 1, 2 , ... ,n, ts . a Iso a d.tstn"b utwn. . F unctwna 1

Example 1.2.7. The Heaviside unit function His defined by H(x)={1,x;:,O O,x(x) = e-lxl 2 E Y(lRn). Proposition 1.2.11.

(i)

~ C

(ii)

~

Y with continuous injection

is dense in Y

(iii)YCLP,l (iv) LP

c Y' c

~p

(1.2.16)

gf is continuous in Y'.

Let f E Y' (JRn ), then the convolution f

* g is defined by

(f*g,lf>) := (f(x), lfl(x)) ,l/> E Y(JRn), where lfl(x) = (g(y), lf>(x+ y)) E Y(lRn). It can be proved that f

( 1.2.17)

* g E Y'(Rn).

Another space, which is used in the development of theory of Hilbert transform, is ~LP(JRn).

It consists of all those COO -functions lf> on JRn such that

Yk(lf>) :=

(k." ioklf>(x)IPdxYIP
(x)dx, }JRn where

a,f3

EN

1/31 = /3 1 + /32 + ... + f3n; so that of3 wa4'( w) is bounded for all a, f3.

0,

It then follows

by the product rule for derivatives and induction on f3 that waof34' is bounded for all Hence

a,f3.

4' E .9'(~11 ).

Next, applying same arguments to (1.3.2), we can show that if 4' E .9'(~11 ) then l/J E Y(~11 ). By uniqueness of the Fourier transform it follows that the Fourier transformation is an isomorphism on .9'(~ 11 ). Now, the Fourier transform §[f] of tempered distribution f E YJI is defined by (§[fJ,cp) := (f,§[cp]) ,l/J E Y.

( 1.4.2)

It specifies an automorphism of the tempered distribution spaceY'. For functions f(t) E

L 1 (~11 ), §[f] coincides with (1.4.I).

Let us define for f E Y' the operator §- 1 [f(t)] = (2n)-n§[f(-t)];

( 1.4.3)

it is also an automorphism of Y' and defines the inverse transformation of§(!): .~-I[§(!)]= §[§- 1(!)] =

f, fEY'.

For f(t) E L 1 (~ 11 ) the formula( 1.4.3) has the form §-I [f] = (2n)-n

r

}!Rn

ei(x,t) f(t)dt.

The following relation holds : §[§[f(t)]] = (2nt f( -t). Example 1.4.2. (i) §[ xo.

Definition 1.8.1. Let f(x) be defined and continuous on domain JR. The formal power series [;;'= 0 an (x- xo)n is said to be an asymptotic power series expansion off, as x-> xo in IR, if the conditions lim {(x-xo)-m[f(x)- Ean(x-xo)n]} =0,m=0,1,2, ...

(1.8.1)

n=O

X->Xo

are satisfied. It can be readily seen that conditions ( 1.8.1) are equivalent to m

f(x)= L,an(x-xot+O(x-xo)m+l,x ->Xo iniR, m=0,1,2, ....

(1.8.2)

n=O

Definition 1.8.2. The formal power series E;;'=oanx-n is said to be an asymptotic power series expansion off, as x

-> oo

if the following equivalent set of conditions are satisfied

lim {?[f(x)x----+oo

Eanx-n]} =

n=O

0, m = 0, 1,2, ... ,

( 1.8.3)

m

f(x) = L,anx-n+O(x-m-l),x->oo,m=0,1,2, .... n=O

( 1.8.4)

An Overview

17

If (1.8.3) or (1.8.4) holds, then we write

r. anX-n ,X---> oo. n=O 00

f(x) rv

( 1.8.5)

In this book we are concerned with asymptotic expansion of integrals. As an example, we consider Laplace transform:

I(x)

= fooo e-xtf(t)dt,

X> 0.

( 1.8.6)

We assume that f(t) is locally absolutely integrable on (0, oo) and that, as t

-->

oo, ( 1.8.7)

for some real number

a.

In order to study l(x) for large x, let us further suppose that, as t

-->

0+,

00

f(t) rv

L

a5 ts+a-l, a> O,t __, 0+. (1.8.8) s=O Then the integral (1.8.6) is absolutely convergent, if x > a and a > 0. Replacing f(t) in (1.8.6) by the expansion (1.8.8) and then integrating term by term we get the asymptotic expansion

=

E

as f'(s

s=O

XS

:a

a) ,x--> oo.

( 1.8.9)

This is the conclusion of the following [92]:

Lemma 1.8.3 (Watson's lemma). In ( 1.8.6) let f(t) be a locally integrable function on

(O,oo ), bounded for finite t and let (1.8.7) and (1.8.8) hold. Then (1.8.9) holds. If we set n-1

f(t)

=

L asts+a-l + fn(t)

(1.8.10)

s=O then ( 1.8.9) can be written as

I(x)

=

n-1

~ a5 L..

s=O

f'(s+ a) xs+a

+ Dn(x),

(1.8.11)

where (1.8.12) is called the remainder term.

18


We can find an estimate to the remainder term [92]. The growth conditions ( 1.8.7) and (1.8.8) suggest that (1.8.13) for some number M11

> 0. The best value of Mn is given by Mn

=

sup lfn(t)tl-n-ae-atl·

(1.8.14)

(O,oo)

The error term in (1.8.12) is bounded by 18n(x)l :(; Mn fooo tn+a-le-(x-a)tdt

= Mn r(n+a) = O(x-a-n). (x- a)n+a

1.8.1

(1.8.15)

The Mellin transform technique

We often use Mellin transform in deriving asymptotic expansion of integrals. We define the Mellin transform of a locally integrable function on (0, oo) by

M[f;z] = fooo tz-lf(t)dt,

(1.8.16)

provided the integral converges. Its domain of analyticity is usually an infinite strip a < Re z < b. The inversion formula for this transform is given by

f(t)

= -2

1 1c+ioo

.

'!Cl

c-ioo

rzM[f;z]dz,

( 1.8.17)

where a < c < b. In asymptotic analysis sometimes we use Abe/limit of a function defined by lim

r= f(t)e-E dt,

E->O+}o

1

(1.8.18)

if this limit exists, in which case we say that fis Abel summable. Iff is absolutely integrable on (O,oo), then Abel limit is simply the integral of f. The improper Riemann integral of

f on (0, oo) is also equal to the Abel limit off [93, p. 197]. However, in the following lemma, we show that the Abel limit may exist even when improper Riemann integral does not converge.

(1.8.19)

An Overview

Proof.

19

We know that

{"" tA.-Ie-ztdt =

lo Replacing z by £

-

ix with

£

f'(~) ,Re z > O,Re z

A> 0.

( 1.8.20)

> 0, gives

lim {"" tA.-Ie-(e-ix)tdt E--+O+}o

=

lim f'(A.) e--+0+ (t:- ix)A.

=

f'(A.) . ( -ix)A.

This proves the lemma.

D

Using Mellin transform and Abel limit, asymptotic expansion of the Fourier transform has been obtained in [91]. A similar technique will be used to obtain asymptotic expansions of wavelet transforms in Chapter 10. 1.8.2

The distributional approach

Distributional technique developed by Wong [93] is often used in deriving asymptotic expansions of certain integral transforms. Assume that

I(x)

= looo f(t)h(xt)dt,

(1.8.21)

and f(t) possesses asymptotic expansion

r. a rs-a ,r-> co, s=O 00

f(t)'"" where 0
+co, s=O where c is real, p

> 0 and 0 < f3

~

( 1.8.24)

1.

Let f be a locally integrable function on (0, co) and let it define a tempered distribution by

(f,l/J) := fooo f(t)l/J(t)dt,l/J E Y(JR).

( 1.8.25)

Let us define

fn,o(t) fn,j+l(t) = -

1

oo

t

fn,j('r)d'r=

= fn(t) (-1 )j+ I .1 1·

1

oo

t

.

(-r-t) 1 fn('r)d-r,

20

for j


= 0, I, 2, ... , n -

O(t-a) as t---->

co.

I. For 0

co and fn,n(t) = O(logt) as t----> 0+.

Therefore, by integration by parts, we find that (1.8.26) The following lemmas give basic identities for derivation of asymptotic expansion off [93, pp. 296-297].

Lemma 1.8.5. ForO< a< I, n;;:: I and q, E Y(O,oo), n-1

n

L

(f,l/') =Las (t.:;:s-a,q,)s=O

Cs (

D(s-1) ,q,)

+ (fn,l/'),

( 1.8.27)

s-1

where

(-1)'

Cs

(1.8.28)

= (s -I)!M[f;s],

M[f;z] is the Mellin transform off(t), or its analytic continuation.

Lemma 1.8.6. If a= I in (1.8.22) then for each integer n;;:: I and for any q, E Y(O,oo), we have

(f,q,) =

n-1

n

s=O

s-1

L as(t.:;:s-l ,l/')- Lds ( D(s-l),q,) + (fn,l/'),

(1.8.29)

where

.

ds = hm

t__,O+

[

fs,s(t)

+ ((-1)'-1 ) s- I .1

]

as- I logt .

(1.8.30)

These lemmas were used by Li and Wong [46] for obtaining asymptotic expansion of convolution ( 1.3.4 ). We shall use these results for deriving asymptotic expansion of the wavelet transform for large values of lbl in Chapter 9.

Chapter 2

The Wavelet Transform on LP

2.1

Introduction

The Fourier transform has been the most useful technique for the frequency analysis of a signal for a long time, but to deal with signals which are not localized in frequency but also in space one needs wavelet transform. The wavelet transform off with respect to the wavelet 1f1 in n dimensions is defined by

t-b (W f)(b,a) =a-n I2 1m f(t)lfl(-)dt, a> 0, bE JR.n a

JRn

=

where Xa(x)

U* Xa)(b),

= a-n/ 21f/( -xfa), provided the integral exists [45, p.

(2.l.l)

(2.1.2) 28].

If f(x) and lfl(x) are both bounded functions of class L 1 then by [6, Theorem 13, p. 24], for

n = I, we can write (2.1.1) in the form:

(Wf)(b,a)

=

al/21oo ~ . = 2 7C -oof(ro)e'brolfl(aw)dw,a > O,b E JR.,

(2.1.3)

where f denotes Fourier transform off defined by ( 1.3.1 ). Iff, 1f1 E L 2 (JR.), then also by Parseval formula, (2.1.3) follows from (2.1.1) [ 12, p. 61]. For more general situations refer to (8.1.9). The form (2.1.3) of the wavelet transform was used by Bochner and Chandrasekharan [6, p.l3] as early as 1949 as a summability integral in the development of the theory of Fourier transform. According to [6, p.l4] we write

S~ (b)=_!_ 1 2n

00

j(ro)eirobK(rojR)dro, R > 0.

(2.1.4)

-00

From (2.1.3) and (2.1.4) it follows that s~ (b)= a- 112(W f)(b,a), 21

(2.1.5)

22


for R = I fa and K( ro) = lif( ro ). Iff E L 1 (IR) and lfl E L 1 (IR), then from (2.1.1) we get (2.1.6)

(cf. [6, lemma 2, p. 15]).

The modified Hilbert transform Ha(f) is defined by (Haf)(x)

= n- 112 r(l -

a)cos(naj2)

1:

which reduces to Hilbert's transform (1.5.1) when

It -xla-l sgn(t- x)f(t)dt

(2.1.7)

a= 0.

If 1 < p < a- 1 and f E U(JR.), then Haf E L', r- 1 = p- 1 - a and there exists a constant M = M(p, a) such that

IIHafllr ~ MIIJIIP

(2.1.8)

[43].

Iff E U(JR.), and g = Haf, then f(x) = - lim n- 1r(l 8---.0+

+ a)cos(naj2)T8(g)(x)

(2.1.9)

for almost all x E Rand also in LP- norm [33], where T8(g)(x) =

r

Jlt-xi-;:,8

It -xl-l-asgn(t -x)g(t)dt.

(2.1.10)

The range of Ha was characterized by Herson and Heywood [33] as follows: A function g E L'(IR) is also in Ha(LP) if and only if IIT8giiP is bounded for all positive 8.

A variant of Ha is Riesz fractional integral operator Ia. 0 I, 0 L' is continuous

and there exists a constant K = K(p, a) such that (2.1.12) If g(x)

= (Iaf)(x),then

f(x) = (I;; 1g)(x) =-lim n- 1r(l- a)sin(na/2) 8--+0

X

r

Jlt-xi-;:,8

It -xl-l-a[g(t)- g(x)]dt

pointwise almost everywhere and also in U-norm [33].

(2.1.13)

23


For Parseval relations involving the operators Ha and Ia we may refer to Theorem 1.1, given below. Let f be defined on ( -oo, oo) and x E ( -oo, oo); then the extended Weyl integral is defined by (2.1.14) and the extended Riemann-Liouville integral is given by

(Raf)(x) = (W; f)(x) := (r(a))- 1 For f

E

J:,., (x-t)a- /(t)dt. 1

(2.1.15)

U(JR.), the following properties ofthe above operators obtained by Kober [44] will

be used in the sequel:

2sin(naj2)Haf = (Wa- w;)J,

(2.1.16)

2cos(naj2)1af = (Wa + W;)J,

(2.1.17)

lxl-aWaf-> f, lxl-aw; f-> f, lxl-aHaf-> Hf in U- norm.

(2.1.18)

Many properties of these operators can be found in [54]. The following generalizations of the operators H, Ha and Ia were given by Okikiolu [52]. For J.l, v E JR. and 0 1, 0 < J.l ~a< 1 (or 0 < a < 1 if J.l = 0), 1/p-1 < v < 1/ p- a, 1/r = 1/p- J.l > 0. Then there exists a constant K = K(p, a, V,J.l)

such that

IIH~~~ (f)llr ~ Kllfllp, IIIt~(f)llr ~ tan(na/2)KIIfllp·

(2.1.22)

Also,for g E L1 (JR.), 1/r+ 1/r' = 1, we have

i: g(t)Ht~(f)(t)dt i: =-

/_: g(t)ItMt)(t)dt where v'

= J.l - v - a.

f(t)Ht)(g)(t)dt,

= /_: f(t)ItJ

(g)(t)dt,

(2.1.23) (2.1.24)

24


Theorem 2.1.2. Let f E LP(JR), p

(i) (ii)

> 1, 1I p- 1 < v < 1I p. Then IIH(v)(f)IIP ~ Ap,vllfllp H(v) { H(v)(f)} = - f

(2.1.25)

a.e.

(2.1.26)

(iii)forgELP'(IR), llp+llp'= 1,

L~ g(t)H(v)(f)(t)dt = - L~ f(t)H(-v)(g)(t)dt

(2.1.27) (2.1.28) (2.1.29)

Now we recall certain results on weighted LP- spaces which are to be used in Section 2.4. Suppose that 0

0 b

as a--> '

lll>n

E~ .

Proof. From [6, p. 100], for R = I/ a, and definition (2.1.1 ), we have I I f t-b SR(b) := 2n-l(r(n/2))n/2 an }JRnf(t)l!'(-a-)dt

I I 2n-l(r(n/2))n/2 an/2 (Wf)(b,a).

Now, the proof follows by an application of Theorem 54 in [6, p. 101 ]. The next theorem gives a result on pointwise approximation in Theorem 2.2.8. Assume that (i) (ii) (iii)

f(x) E

U'(~n),

p

>I

lj/(ro) is radial, and ij/(w) ~ 0

(Z~)n fJRn lj/(ro)dro =I

~n.

(2.2.9)

The Wavelet Transform on U

(iv)

29

ijl(co) = 0 Ccol~+e), e > 0, as Icol--+ oo,

then

l

gb(t)dt = o(t), as t---> 0

implies

as a---> 0. Proof. See [6, p. 103]. 2.3

Wavelet transform, Hilbert transform and Fractional integrals

In this section we consider the general form of the wavelet transform studied by Kaiser [41]:

1

t- b dt (Wf)(b,a)= -oof(t)l!'(-a-)aP' p>O, 00

(2.3.1)

and obtain its relations with the Hilbert transform H, fractional integral operators Ha and

Ia, 0 O.

Therefore, by relation (2.3.12),

r

1

.

Example 2.3.2. Consider the Mexican hat wavelet defined by

which is an even wavelet. Then for 0
w;(~ x ~+), r= 2- p- a, r- 1 = and f(t)

=-

. atan(naj2) hm 2 nAa

s~o+

h

lb-tl~o

lb

p- 1 -

a,

1 ~o= !Y+I db [(W f)(b,a)- (W f)(t,a)]a-Yda

- t

o

(2.3.19)

pointwise almost everywhere and also in LP-norm. The range of the wavelet transform can be characterized using the characterization of the range of Ha (and also of Ia) [33]. Theorem 2.3.9. Let p, a, f and 1/f satisfy conditions of Theorem 2.3. 7 when 1/f is odd and

that of Theorem 2.3.8 when 1/f is even. Let F(b,a) be any function in Wy(~ x ~+), r- 1 =

= (W f)(b,a)for some f E LP(~) if and only ifiiTsOasa-->0+,(11' odd). (2.3.22)

Proof. From (2.3.8),(2.3.9) and (2.l.l7) we have

lbl-a2))(b,a)aP- 2 da =loco (WF)(O,a)aP- 2 da

(ii)

-loco (W F)(tf>2(b),a)aP- 2 da. Proof. (i) Using (2.3.12) and (2.3.24) we have

loco W (F( 1/11)) (b, a)aP- 2 da =AonH(F( 1/JI) )(b, a) =AotrH(F(x); 1/11 (b)) =loco (W F)( 1/JI (b),a)aP- 2 da; from which (2.3.28) follows. (ii)

From (2.3.12) and (2.3.26) it follows that

loco (W F)( tf>2(b),a)aP- 2 da =AotrH(F(tf>2(x));b) =

AotrH {F(x);O} -AotrH {F(x); tf>2(b)}

=loco [(WF)(O,a)- (WF)(tf>2(b),a)]aP- 2 da. This yields (2.3.29).

(2.3.29)

37


2.4

Wavelet transform on weighted U- spaces

Let w be the weight function satisfying (2.1.31) and U(wP), 1 ~ p 0 independent of o > 0 such that

where y is given by (2.3.8)- (2.3.9).

2.5

Boundedness of the wavelet transform on LP with different weights

In this section, instead of using relation (2.3.9), we derive an inequality between the integral of the wavelet transform and the operator Ia and use (2.1.40) to find a boundedness result for the wavelet transform. We have for y E JR., l!ooo(Wf)(b,a)a-Ydal

~

foooi(Wf)(b,a)la-Yda

C

~ fooo a-P-r (I:if(t) lfl ~b) Idt) da (2.5.1)

(2.5.2) The above change in order of integration can be justified as follows. We see that

I:

jt(t)lfl

C~b) I =I: dt

~

jt(t)[w(t)] 11Plfl

llfllp,w lllflb;allp',w-lfp ·

C~b)

[w(t)t 11PI dt

40


But, for I

=

(l

< p < oo,

oo

[w(t)]

_'

P fp

(

Jt-bJ)(a-l)p' ( 1 + Jt-bJ)(l-a)p'l (t-b)lp' dt )ljp' 1Jf -

I +-a

. -oo

a

00

a

)

::,:; QE j_oo[w(t)rP'!r (I+ Jtl)(a-I)p' dt (

!jp'
0 such that (2.5.5)

and (2.5.6) then by (2.1.17),

l"'

I(W f)(b,a)l a-Yda =

~ B[r(a)Wa(lfl)(b) + r(a)W;(Ifl)(b)J

Br( a)2cos(nai2)Ia(lfl)(b).

(2.5.7)

Then using Theorem 2.1.7 we have the following:

Theorem 2.5.1. Let w be a nonnegative weight function on a set of positive measure satisfying (2.1.40)-(2.1.41). Let 0 0 such that

IIWfllq,a,v ~ C(a) llfllp.w·

(2.5.8)

Applying Theorem 2.1.8 to relation (2.3.9) we get the following boundedness result.

Theorem 2.5.2. Let p, r > 1 and 0 < a

< I with 1I p + I I r = I + a.

Let f E LP(~) and v E U(~). Let 1J1 E u' (~)be an even function which satisfies (2.3.7). Then there exists a sharp constant C( a, p) such that

where y = 2 - p - a

2.6

Wavelet transform and operators H(v), Ht~ and It~

In this section we study wavelet transform of ltlv f(t), where v E ~and fEU(~). At first we obtain relations between W(ltlv f), H(v)(f),

Ht~(f) and ~~~~(!).These relations

are used to investigate approximation properties of the wavelet transform and to establish inversion and Parseval formulae. Some of the results of this section are generalizations of those contained in Section 2.3.

42


Proceeding as in (2.3.2) we find that for any v E JR.,

~a= W(ltlv f)(b,a)a-Yda =~a= a-P-r (l~ f(t)ltlvll' C~ b )dr) da. The t-integral is absolutely convergent iff E U(IR.) and ltlvll'(!~h) E u'(JR.), l / p 1 = l. But, as in (2.4.6) we have, for some r

(2.6.1)

1/p+

> 0,

whereO < v+ 1/p' < r, Q-r

= supi(I + ltlrll'(t)l < oo

(2.6.2)

t

and E' a constant similar to that occuring in (2.4.8). Also, from (2.3.4), the a-integral is absolutely convergent if lJI( ±t)tP+r- 2 E L 1( 0, oo). Then the order of integration in (2.6.1) can be interchanged and for 2 - p - a =

r we can write

~a= W(ltlv f(t))(b,a)a-Yda = r(a)Wa(ltlv f)(b) ~a= lJI(t)t-adt +r( a)W~ (ltlv f)(b)

~a= lJI( -t)t-adt

= 2Aar(a) sin(naj2)Ha(ltlv!)(b),

if lJf is odd

(2.6.3)

= 2Aar(a)cos(naj2)1a(ltlv!)(b),

if lJf is even,

(2.6.4)

where A a is given by (2.3. 7). Therefore, by (2.1.20)-(2.1.21 ),

lblll-v-a ~a= W(ltlv f)(b,a)a-Yda

= 2Aar(a)sin(naj2)HtkU)(b) (II' odd)

(2.6.5)

= 2Aar(a)cos(naj2)1~~1U)(b) (II' even).

(2.6.6)

Next, as in (2.3.12) we find that, for

a = 0,

The Wavelet Transform on U

43

so that lbl-v fo'"" W(ltlv f)(b,a)aP- 2da = AonH(v)(f)(b).

(2.6.7)

Now, applying Theorem 2.1.1 to relations (2.6.5)-(2.6.6) we get the following.

Theorem2.6.l.Letp>1,

0max( I - a, v + 1/ p').Then

lbl-v fo"" W(ltlv f)(b,a)a-Yda- fliP---> 0

as a---> 0+ (lfl even),

(2.6.14)

(2.6.15) where t.(ll(a) = 2Aar(a)cos(anj2) and t.
on JR. belongs to Fp if and only if (2.7.1) Then Fp, 1 ::::;; p < oo, is a locally convex, Hausdorff and sequentially complete topological vector space [49]. Another space useful for the study of wavelet transform is the space G~.v(IR. x JR.+), J1, v E JR.,

1 ::::;; p
). P Aon

(2.7.8)

W(Ok)(b,a)aP- 2da); Ok(t) = (tdjdt/q>(t) = - -1-H ( lo Aon so that by (1.5.2) and (2.7.5), A.{(q>)

~ ACp II on

kr

In view of (2.7.6) and (2.7.8) we conclude that W is a one-to-one mapping from Fp(lR) onto a;l,p-l(JR X JR.+)· Therefore, w- 1 is defined on a;Lp- 1(1R X JR.+). Since w- 1wq, = 1/> for all If> E Fpc LP, it follows that

w- 1 is given by (2.3.14).

Now, we define the wavelet transform W'T of generalized function T belonging to (G;i,p- 1 ) 1 , the dual of G; 1·p-i (JR. X JR.+) by

(W'T,q>) := (T,Wq>),

(2.7.9)

46

where 1/J


E

Fp(IR.).

Using duality arguments we can prove Theorem 2. 7 .2. Let T E (

c; 1,p- 1)',

I

< p < oo and lfl E .9' (JR.) satisfy (2.3. 10), then the

generalized wavelet transform defined by (2.7.9) is a one -one continuous linear map of

(c; 1,p- 1)' onto F~. Next, we assume that 0 < 2- p- r

=a
0. Hence (2.7.12)

where J1 = I - y- p and v = p - 1. From this we conclude that for 0 < 2 - p - y =

a < I, W is a continuous, linear map from

Fp into G~'v. Therefore, as in (2.7.9), we can define generalized wavelet transform ( G~,v (JR.

X

w' T

of T E

JR.+) )

1•

Theorem 2.7.3. Let f E (G~,v (JR. x JR.+))', 1 < p

< oo and 0 < 2- p- y =a< 1. Let

lfl E Y'(IR.) satisfy (2.3.7). Then W'f defined by (2.7.9) is a one-one, continuous, linear map from (G~.v)' into F~.

Following the above technique and the distributional analysis contained in [55] the results of Section 2.4 can also be extended to distributions.

47


2.8 The wavelet transform on LP(I~n) In this section we extend some of the results of Section 2.3 to n-dimensions.

For

this purpose we assume that a= (aJ,az,a3,···,an), a;> 0, i = 1,2, ... ,n and b =

(b h bz, b3, ... , bn) E ~n. Define the wavelet transform by (2.8.1)

(Wy,f)(b,a) = { f(t)lf/b;a(t)dt,

}JRn

where _

lf/b;a(t)-a

(t1-b1 tz-bz tn-bn) If/ - - , - - , ... , - - ' a1 az an

-p

(2.8.2)

and -p a - p-- a -p 1 . .. an .

This transformation with p = 1 has been studied by [74]. Next, we recall the definition and properties of the Hilbert transform inn-dimensions [53, pp. 152-153]. Let f E LP(~n), p > 1, then its Hilbert transform (Hf)(x) defined by

(Hf)(x)=n-nlim~axe;~O+ 1EIIi

{ . On ~(t~ ·)dt Jlr,-x,l>e, l=i t, x,

(2.8.3)

exists a.e and Hf E LP(~n). There exists a constant Cp > 0 such that (2.8.4) Moreover, for all f E LP(~n),

(H 2 f)(x)

=

(-It f(x)

(2.8.5)

in LP(~n) and a.e. Now, multiplying both side of (2.8.1) by a-Y = a~Yl .a~Y2 · · · a;;Yn, with Y1 =

}'2 = · · · Yn =

2 - p and integrating with respect to a = (a 1, az, ... , an) we get

{ (W f)(b,a)aP- 2 da

JJR"+

=

r

lrR"+

f(b+u) du u

r

= { f(b+u)du { lfl(':!_)a- 2da }JRn

lfl(t)dt+

lrR"+

J'iR"+

r

f(b-u) du

lrR"+

u

a

r

lfl(-t)dt.

lrR"+

If we assume that

r

}'!Rn

lfl(t)dt

= o,

(2.8.6)

then for

O=rfAn= { lfl(t)dt=- { lfl(-t)dt,

lrR"+

lrR"+

(2.8.7)

48


we have ( (Wf)(b,a)aP- 2da=An(f

}~!'+-

}~"+

=An p.v. (

f(b+u)du- ( f(b-u)du) u lJR"+ u f(tb) dt,

lJR"+ t-

where t- b = (t,- b, )(t2- b2) ... (tn- bn) and p.v. is taken as in (2.8.3). This gives the desired relation between n-dimensional wavelet transform and Hilbert transform: (2.8.8) Using (2.8.5) and (2.8.8) and proceeding as in Section 2.3.1, we arrive at the following inversion theorem. Theorem 2.8.1. Assume that f E LP(JR.n), 1 < p

Let II' E

u' (IR.n), 1I p + 1I p' =

< oo.

1, be a basic wavelet which satisfies (2.8.6).

Then

(2.8.9) in LP- norm and a.e.. Remark 2.8.2. It will be worth while to investigate other results of previous sections to n

dimensions and extend to distributions.

Chapter 3

Composition of Wavelet Transforms

3.1

Introduction

In this chapter we consider a variant of (2.1.1) and define the wavelet transform of tf> with respect to the wavelet II' by

(Wtf>)(b,a)

=

f

}JRn

1/>(t)l/f((t-b)/a)dt/an,

(3.1.1)

provided the integral exists, where bE ~n and a> 0. If tf> E L2 (~n) and II' E L2 (~n), then using Parseval formula for Fourier transforms we can write the wavelet transform in Fourier space in the following form ([36], p.9):

W(b,a)

= (Wtf>)(b,a) = (2n)-n f ei(lO,b)fil(aro)~(ro)dro. }JRn

(3.1.2)

This form of the wavelet transform is very similar to that of a pseudo-differential operator with symbol a(a, ro)

= fi/(aro).

Hence a theory of the wavelet transform can be developed

analogous to that of a pseudo-differential operator [90]. Assuming that 1/f E .9"(~n) we investigate continuity of the wavelet transform (3.1.2). We define the composition (product) W1 o W2 of two wavelet transforms W1 and W2, which turns out to be another wavelet transform of three variables. The continuity of W1 o W2 is also investigated and a reconstruction formula is obtained. Certain weighted Lebesgue and Sobolev spaces are defined. Results on the boundedness of the wavelet transform in certain weighted Sobolev norms are obtained. Corresponding results for the composition W1 o W2 are also given. 49

so


3.2 The wavelet transform of tempered distributions In this section we need the test function space Y(IR.n x JR.+) defined to the space of all functions c/J E C"(IR.n x JR.+) such that for .e, k E No and

Ye,a,k,f3(c!J)

laeba(ajaalD~c/J(b,a)i < oo,

sup

=

a, {3 E N0, (3.2.1)

(h,a)ElR" xlR+ l+lal~k+l/31

Clearly, the Schwartz space Y(IR.n x JR.+) is contained in .9"(1R.n x JR.+). Theorem 3.2.1. Let lfl E .9"(1R.n). Then the wavelet transform (Wc!J)(b,a) is a continuous

linear map of Y(IR.n) into Y(IR.n x JR.+). Proof. For f,k E No and a,{3 E N0, we have after differentiation and integration by parts,

aeba(a jda)kD~Wc/J(b,a) =

(2n)-nbaD~ .~n ei(b.ro)~(ro) { ae(a jJa/iji(aro) }dro

= (2n)-nba { ei(b.rol(iw)f3~(ro)ie+k (w-e { ei(ro,ay)(D )e[(ro,y/lfl(y)]dy) dro }]Rn

= (2n)-nba x

}]Rn

Y

r ei(b,ro)jlf31+f+kwf3-e~(ro)

}]Rn

(lrn ei(ro,ay)D~ [ IA-I=k L (k!jA,!)roA.llfl(y)l dy) dro lR"

=

(2n)-nif+k+lf3+al fnei(b,ro)

JR

x

c;~a

(a) (Dw)a-8(wf3+A.-e~(ro)) 8

(k.n (iay)c;(Dy)e[llfl(y)]ei(ro,ay)dy) dro

= (ln)-nie+k+lf3+a+281

r" ei(b.w) r, (k!JA,!) r,

JJR X

L (k!JA,!) L IA-I=k

IA-I=k

(a) 00 -8v~-8(wf3+A.-e~(w))

Ci~a 8

(k.n (Dy) 8i(Dy)f(/lfl(y))ei(ro,ay)dy) dro

=

(ln)-n/+k+I+I/3+a+2Cil

L (k!JA,!)L, (a) L (a

IA-I=k X

c;

8

P

-8)

P

r ei(b.ro)A({3,.e,;.,,p)wf3+A.-f-p-CiD~-Ci-p~(ro)

}]Rn

x (

L ( 8 ) A'( 8, y) frn"i-rv;+Ci-Y(/lfl(y))ei(ro,ay)dy) dro.

r~c;

r

lR

51


1/31 + k ~ Ia I + t', we can write laeba( a I da)k D~ q>(b, a) (W q,) (b, a) I

Therefore, for

(a)o

L. (k!/A,!) L.

~ (2n)-n

8~a

IA.I=k

xA(j3, t',A,, p )A' ( X

X

L.

p~a-8

(a -o) L. (o) P

r~8

L.

( + o-y) t'

Y -r~e+8-y

-r

o, y)A" (A,, -r, If!)

r (1 + lroi)I.BI+k-f-lpl-181+n+IID~-8-p ~( (0) ldro/( 1 + lrol)n+

}!Rn

k..

(1

I

+ lyl)k+l81-1-rl-lrl+n+IIDf+8-y+-rlfi(Y)Idy/(1 + IY!t+l;

so that

Ye.y.k,,B(WI/>)

~ L (k!/A.!) L (a) L 8~a

IA.I=k

X

L

(

p~a-8

(a-o) L (o) P

y~8

Y

~

t'+o-r)

B( a, J3' y, -r, A, o,p, t', n)Y!.BI+k-f-lpl-181+n+l,a-8-p ( q,)

-r

-r~e+8-r

o

x Yk+l81-lrl-l-rl+n+ 1/+8+-r-y( If!)·

Thus Wlf>(b,a) E 9(~n x ~+),and from the above inequality the continuity of WI/) also follows. In view of the above theorem the generalized wavelet transform W'T ofT E 9', the dual of 9 (~n x ~+), can be defined by (3.2.2) Using duality arguments we have

Theorem 3.2.2. The generalized wavelet transform W' : 9'

--->

9' is linear and continu-

ous. We can also analyse the wavelet transform by imposing a condition on the Fourier transform of the wavelet 1f1 as follows: Assume that lj) E Coo(~n) such that (3.2.3) where m E set of all

~.

0

~

p

~

1,

a

E N

0. Let us define the function space 9

1 (~n

x

~+)

to be

c= -functions q, on ~n x ~+ such that fort', k E No and a, J3 E N0, Ye,a,k,,B(I/>) =

sup (b,a)EIR"x!R+ l+lal~k+I.BI l+lal~-m

laeba(a;aa)kD~If>(b,a)l < oo.

(3.2.4)


52

Theorem 3.2.3. Assume that the wavelet lfl satisfies (3.2.3 ). Then the wavelet transform

.9", (JRn x JR+) and the generalized wavelet transform is a continuous linear map of .9"{ (JRn x JR+) into .9"' (JRn). W is a continuous linear map of Y(JRn) into

Proof. Proceeding as in the proof of Theorem 3.2.1 we can write Jaeba(() jJa)kD~ W !f1(b,a)

J

L (k!/A.!) L (a) j[D~D~i{/(u)]u=aroJalrllwiiA-1

::_;; (2n)-naffm IR"

x

E

8.;;;a-r

::.:;; (2n)-n

IA-I=k

(a-

y.;;;a

r) A(/3,

c5

c5)1rollf3- 8 11D~-r- 8 ;p(ro)ldro

E (k!/A.!) L E IA-I=k

r

y.;;;a8.;;;a-r

(a) Y

(a-r)A(f3,c5)alrl+f c5

r lroJIA-1+1/3-81(1 +aJroJ)m-p(lri+IA-IlJD~-Y- 8 ($(ro)ldro ::.:;; (2n)-n L L L (k!/A.!) (a) (a-r)A(f3,c5) X

./JRn

IA-I=kr.;;;a8.;;;a-r

Y

c5

r (I +aJroi)IYI+f+m-p(lrl+kllwlk+lf31-181-1rHID~-y-8($(ro)Jdro ::.:;; (2n)-n L L L (k!/A.!) (a) (a-r)A(f3,c5) X

./JRn


X

Y

c5

r (l + lwl)k+lf31-181-lrl-f+n+IID~-y-8($(ro)Jdro(I + lrol)-n-1

./JRn

for lrl +t'+m- p(lrl +k) < 0. Therefore, form< -t' -Ia I and t'+ Ia I::.:;; k+ 1/31, we have

Ja'ba(d/da)kD~W!f1(b,a)l::.:;; (2n)-n E E E

(k!/A.!)


(a) (a- r) Y

c5

xA(/3' c5, n)Yk+l/31+181-lrl-f+n+ l.a-y-8( ($). From this we conclude that the wavelet transform is a continuous linear map of Y(JRn) into ,9"1

(JRn

X

JR+).

As in Theorem 3.2.1 we define the generalized wavelet transform W' ofT E .9"{, the dual of .9"1(JRn x JR+ ), by (3.2.2) and get the second part of the theorem.

3.3

Composition of wavelet transforms

Let W1 and W2 be two continuous wavelet transforms defined as follows: (W,q,)(b,a)=(2n)-n { ei(b,ro)i/f,(aw)($(w)dw,

./JRn

bEJRn, aElR+

(3.3.1)

53


and (3.3.2) provided the integrals exist. Then their composition(product) Wt o W2 is defined by

(Wlf>)(b,a,c)

= (Wt o W2)l/>(b,a,c) = (2n)-n { ei(b,lO)Vtt(aco)(~d(W2l/>))(co,c)dco }JRn

(3.3.3)

(3.3.4) where

~d

denotes the Fourier transformation with respect to the variable d and

x(a,c,co)

=

(3.3.5)

Vtt(aro) Vt2(cro),

provided the integrals involved are convergent. For instance, W1 o W2 is well defined when if> E L 2 (1R.n),l/fi E L 1(JR.n) and 11'2 E L 2 (JR.n). From (3.3.4) it is obvious that the composition

Wt o W2 is a wavelet transform of three variables mapping from IR.n into IR.n x JR.+ x JR.+. Now, we impose certain conditions on

Vfl, o/2 and if so that (3.3.4) may be meaningful and

of sufficient use. Assume that if> E Y'(IR.n), so that

if E Y'(JR.n) and (3.3.6)

and (3.3.7) where Ca, Ha are constants and m 1 , m 2 are real numbers. As in Section 3.2, let Y'(IR.n x

JR.+ x JR.+) consist of all C" -functions ( b, a, c) such that

1111 =

sup

la'c1 ba(a;aa/(a;ac)PD~(b,a,c)l < oo,

(3.3.8)

(b,a.c)EJRn xiR+ xJR+)

for

Ia I+t'+t::.::; 1/31 +k+ p, t'+ lal +mt::.::; 0

and t + Ia I+m2::.::; 0.

Theorem 3.3.1. The composition operator Wt 0 w2: Y'(IR.n)--> Y'(IR.n

and continuous

X

JR.+

(3.3.9)

X

JR.+) is linear

54


Proof. Proceeding as in the proof of Theorem 3.2.1, from (3.3.3) for f,t,k,p E No and

a,fj

E

N0, we have

laf d ba(a 1aa)k(a I ac)P D~ (W l/J )(b,a,c) I

= (2n)-n lafc ba 1

~ (2n)-naf c1

k.n (iw)f3ei(b,co) {(a 1aa/ VII (aw)(alac)P Vl2(cw)4}(w) }drol

kn I

(a I aw)a [ {(a I aa)k VII (aw) (a I ac)P Vl2(cw)} wf3 4}( ro)] Idw

~ (2n)-naf c JJRr Iy,;a L (a) (a I aw)Y {(a I aa/ VII (aw) (a lac)P Vl2(cw)} Y 1

n

x l(alaw)Y- 8(a lac)PV12(cw)l

v,;~-r (a~ Y) A(fj, v) lwf3-v(alaw)a-r-v4}(w)l dw.

Now, using estimates (3.3.6) and (3.3.7), we see that the last expression is bounded by

(2n)-n r~

X

(a) y

lwlk+p-f-lrl

(r) ftr

L

o

v,;a-y

f (1+alrol)t+l81(1+clroiY+Irl-l81 C(y,o,p,k) }JRn

(aV

y)

(1 +alrol)-m' (1 +clrol)-mz

A(fj' v) Iwf3-v (a I aw )a-y-v 4' ((J)) Idw

~(2n)-n y,;a L (a) L (y)C(y,o,p,k) v,;a-y L (a-y)A(fj,v) Y 8,;y 0 V ~ (2n)-n y,;a L (a) L (y)C(y,o,p,k) v,;a-y L (a-y)A(fj,v)YA.,Jl(4}(w)) Y 8,;y 0 V (3.3.10)

IPI-Ivl-lrl +k+ p -.e -t +n+ 1, J.l =a- y- v, t+ Ia! +m2 ~ 0 and t+ Ia! +f+t ~ IPI +k+ p.

where It=

From (3.3.1 0) the assertion follows.

f+

lal +m1

~ 0,

55


We can define the generalized composition operator (W1 o W2 )' of any T E Y'' (JRn x JR+ x

JR+) by

Again, by duality arguments, we have

Theorem 3.3.2. The generalized composition operator

is linear and continuous. Next, we obtain a reconstruction formula for the product wavelet transform.

Assumption 3.3.3. [Admissibility condition] Let 1/IJ E L2 (JRn), 1112 E L 1(1Rn) be such that there exists a positive constant C'l'1,'lfz

< oo and, for ~ almost everywhere on JRn (3.3.11)

Theorem 3.3.4. Let lflt E L2 (1Rn), lf/2 E L1(1Rn) and c/J E L2 (1Rn) which define the compo-

sition operator CW c/J )(b,a,c) by means of(3.3.3). Then

f

}JRn

Jor= Jor=(Wf)(b,a,c)Wg(b,a,c)dbdadc/(ac)

=C'If1,'1f2 (!,g),

(3.3.12)

for all j,g E L2 (1Rn). Moreover, iff E L2 (1Rn) is continuous at x E JRn, then f(x) = - 1 - { r= r= (W f)(b,a,c)U(x- b,a,c)dbdadc/(ac), c'l'l ,'lfz }JRn lo lo

(3.3.13)

where, (3.3.14)

Proof. Let us write

and

Then, using unitary property of the Fourier transform, we have

-

-

dade ~lo~ (W f)(b,a,c)(W g)(b,a,c)db-llo a c 0 JRn 0

56


This proves (3.3.12). Now, to prove (3.3.13), we set g(t) = 8a(t- x), x E JR., where 8a(x) is the Gaussian function defined by

8a(x) with 8;;(ro)

=

= T 1(na)-! exp( -x2 /(4a)), a> 0,

e-aro 2 • Then, using the approximation property of the Gaussian func-

tion([12], Theorem 2.8, p. 29), we get lim

- f)(b,a,c) (W 1m looco loco o

a-o+ IR"

(loco e-zbrofi/1 . (aco)ljh(cco)e'roxe-aro . dco 2

o

)

da-de db a

This yields (3.3.13).

3.4

Weighted Sobolev Spaces

In order to study singularities at the origin we need the following function spaces. A measurable function lfl on IR.n is said to belong to the weighted space L~l(JR.n), 1 ~ p

< oo, s E JR. and a E IR.n, if

lllfiiiL~P(JR") = Clearly, Lg'P(JR.n) = U(!R.n). Let us set

(k.n I(I+ !u! )-sl ualf!(u) IP du) 2

2

l/p

< oo.

c


57

Then, a tempered distribution f E Y'(JR.n) is said to belong to the weighted Sobolev space H~P(JR.n), I ~ p

< oo, s E JR., a E IR.n, if

d;)

11/IIHt/{IR") = (k.. ILs,af(;)IP

l/p

< 00 •

Clearly, H~·P(JR.n) = W·P(JR.n), the well-known Bessel potential [90]. Using the well-known property of the Fourier transform for a, ann-tuple of non-negative integers, we have

11/IIH~P(JR") = (k.n 1§- 1(1 + 1;1 2 )-s/ 2§(Da f)(;)lp d;) l/p. Therefore, Hd_P C Hs,p for

a

E N

0. For p = 2, Plancherel's theorem gives

11/IIH~{IR")= (k."(I+I;I 2 )-· ·I;aj(;)l 2 1

d;y/ 0), ::::; (2n) IIC/JIIHs(JR) llli'IILo(IR) (s::::; 0). For f3

(ii)

= (0,0), IIWC/JIIH'"(JR2)::::; (2n) IIC/JIIH~ 112 (JR) llli'IIL2(JR) (s > 0), ::::; (2n) IIC/JIIH~ 112 (JR) llli'IIL0(JR)

Theorem 3.5.4. Let

f3I

q,, lJII, 11'2 E ..9"(~).

Then for s E ~.

(s::::; 0).

f3 = (/31 ,f3z,/33)

E IR3 and 8

=

+ f3z + /33- 1, we have

Proof. The proof is similar to that of Theorem 3.5.3, where Lemma 3.5.2 is to be applied instead of Lemma 3.5.1.

3.6

Abelian theorems for wavelet transform of functions

Initial and final value theorems for wavelet transform (3.1.2) with n=l are given in this section. In what follows we assume that

lj/(m) = O(JroJil), Jml----> 0

(3.6.1)

and set (3.6.2)

+ 2,

> 0. Assume that JroJ 1-17lj/( co) E L 1 (IR), Jlj/( m)J ::::; M, M > 0, and ~ (co) E L 1 ( 8, oo), \f8 > 0. If Theorem 3.6.1. Let 2 < 1] < J1

J1

(3.6.3) then lim a2 -11W(b,a) = aH(T]).

a->=

(3.6.4)

60


Proof. Using (3.6.1) we can write

ia 2 -11W(b,a)- aH(1J)I

=

la

2 -17 (2n)- 1

l~ eibwfji(am)~(m)dm- a l~ fji(am)(alml) 1- 11 adml

=a \l~ [(2nt 1 eihro~(m)lmi-I+TJ- a J (almi) 1-1J fji(am)dm\

~a

sup lrol 2, keeping D fixed the second tenn in (3.6.5) can be made less than t::/2 for all sufficiently large a. Theorem 3.6.2. Let 2 < 1J < J1+2, J1 > 0. Imill~( co) E L 1(-X ,X), VX

Assume that lmi 1-1Jlji(m)

E

L1(IR) and

> 0. If

lim (2n)- 1 eibro~(m)lml-l+1J lrol_,oo

=a,

(3.6.6)

then Iima 2 -11W(b,a) = aH(1J).

(3.6.7)

a--->0

Proof. As in the proof of the previous theorem, for X> 0, we have ia 2 -11W(b,a)- aH(1J) I

=a

r

Jlroi<X

1(2n)-leibro~(m)lml-l+1]_ al (alml) 1- 11 1lfi(am)ldm

~ a2 -11

1:

+

1(2n)- 1 eibro~(m)lmi-I+1J- al Joo

sup IWI>X

I (2n)- 1 eibro~(m)lmi-I+1J- allml 1- 17 1lfi(am)ldm

-00

lml 1- 11 llfi(m)ldm,

61


e is an arbitrary small positive number. In view of the asymptotic behaviour (3.6.1 ), there exists a constant A > 0 such that Ii/f(am) I ~A (al ml)ll. where

Hence

la 2 -11W(b,a)- aH(1J)I

~ Aa2 -11+Jl /_: I(2n) -I eibwc/)( m)- al ml 1- 17 11mllldro +

sup 1(2n)- 1eibroc/)(m)lml- 1+17 IWI>X

-

alj= lml

1- 17

1i/f(m)ldm.

(3.6.8)

-=

Since both the integrals on the right-hand side of (3.6.8) are convergent, and second term is independent of a, for given e > 0, the second term can be made less than ej2 by choosing X sufficiently large. Then there will exist B > 0 such that when 1J < 2 + J1 the first term is less than ej2 for 0 0. For h > 0, we have

k[W(b,a+h)- W(b,a)]- 2~ (cl)(m), -Jaeibw~) =

2~ ( c/)( m),eibw { ~ ( i/f[(a + h)m]- i/f(am)) -

:a i/f(am)}).

We need to show that

eibw{~(i/f[(a+h)mJ-~)- :ai/f(am)}->Oin Now, denoting

(fro

1(t)k

r

Y(JR:) as h--->0.

i/f(am) by i/lr(am ), we have

(a~)

m

[eibw { ~ ( i/f[(a +h)m]- i/f(am)) _ :a i/f(am)} J1

62


Thus . W(b,a+h)- W(b,a) _ I \~( ) d ibro-=---()) I1m h - t/J m , :~e 1f1 am . 2n aa

h--->0

Similarly, we can prove the differentiability with respect to the variable b, and in general we have (3.7.2). Next we obtain asymptotic orders ofW(b,a).

Theorem 3.7.2. Let W(b,a) be the wavelet transform of~ E S"''(IR) defined by (3.7.1). Then, fork large, we have W(b,a)

= O(a- 2klblk),a ---7 0; = O(ak),a

---7 oo;

=

O(a-k(I +a 2 )k), lbl---7 0;

=

O(a- 2k(I

+a2 llblk), lbl---7

00 •

Proof. By the boundedness property of generalized functions ([97], p.lll) there exist a constant C > 0 and a non-negative integer k depending on ~such that

IW(b,a)l

~ Cs~p 1(1 + m

2/

(

}m

r{

eibroi/f(am)} I


63

z = aro

~ C stato (;) ( ~) lblk-s~r,s (o/(z)) as-2r

~ c'fo (~)a- 2r(a+lbl/ = C'(I +a- 2)k(a+ lbl)k,

(3.7.3)

from which the result follows. We shall need the following lemma for obtaining Abelian theorems for the distributional wavelet transform. In this section we assume that (3.7.4) for some real number fl.

Theorem 3. 7.3. Let ll' E .7 (~) and cfJ E .7' (~) be a distribution of compact support in ~. Then

is a smooth function on

~

x ~+ and satisfies

W(b,a) = O(a.U(l +a+ lbl/), a----> 0, kEN.

(3.7.5)

Proof. Let ll' be in .7(~). then VIE .7(~) and as a function of ro, eibroo/(aro) E 6"(1R), the space of all C"'- functions on Let cfJ E .7'(~), then

~.

iii E .7'(~). Moreover, assume that iii is of compact support K C R

Let A(ro) E @(lR), the space of all coo -functions of compact support, such that A(ro)

=I

in a neighborhood of K. Therefore, W(b,a) =

2~ (iii(ro),eibroo/(aro)) = 2~ (iii(ro),A(ro)eibroo/(aro)).

Then by Theorem 3.7.1, W(b,a) is infinitely differentiable with respect to the variables b and a. By boundedness property, as used in the proof of Theorem 3.7.2, we have IW(b,a)l

= 2~ l(iii(ro),A(ro)eibroo/(aro)

)I

64


~ Cm~x !~~ ~D~ [.:t(co)eibroijl(aco)] I

t(

~ Cm~x roEKn=O sup :S

~D~-n;qco)D~ (eibroi/f(aco)) I

r)

n

Cm~x ~~~nt (:) st (:) ID~-n;t(co)il(ibt-s (eibroD~i/f(aco)) I

~ C'max sup t r

t ( r) (n) ID~-n.:t(co)llbln-sas+.Uicoi.U n

roEK n=Os=O

~ C"m~x t

n=O

( t r) n

S

(n) lbln-sas+.U

s=O

S

~ C" m~x nt (:) (a+ lblta.U = C"max(l +a+lbiYa.u, r

(3.7.6)

where C" is a positive constant. This gives the required result. The initial value theorem for the distributional wavelet transform (3. 7.1) is given by:

Theorem 3.7.4. Assume that (ji E 9' can be decomposed into

(ji =

C/11

+ C/J-2,

where

C/11

is an

ordinary function and C/12 E g' (IR \ 0) is of order k. Let the real numbers Jl and T/ be such that 2 + k < T/ < 2 + Jl. Finally, assume that

lcol 1-17 i/1( co) E L 1 (IR) and cp 1(co) E L1 ( D, oo )\fD > 0. IJW(b,a) is the distributional wavelet transform of (ji defined by (3. 7.1 ), then lim a2-11W(b,a) = H(T/) lim _!_{ji( co) lcol-l+1J.

a~=

lroi~O

(3.7.7)

2n

Proof. By Theorems 3.7.1 and 3.7.2,

W2(b,a)

=

1 2 7r ( C/J2(co),eibroijl(aco))

is an infinitely differentiable function on IR x IR+ and W2(b,a) = O(ak), as a--->

oo.

Hence

there exists a constant A > 0 such that

Also, since the support of C/J-2 E g' (IR \ {0}) is a compact subset of IR \ {0}, lim eibrol/J2(co)co-I+1J = 0. ro~o

The conclusion of the theorem follows by an application of Theorem 3.6.1 with {ji(co) replaced by

C/11

(co).

65


Final value theorem for the distributional wavelet transform is the following:

Theorem 3.7.5. Let 2 < TJ < 2 + Jl. Assume that iii E Y' can be decomposed into iii= t/J1

+ t/Jz,

tPz

E

where t/J1 is an ordinary function satisfying lmiJltfJI (co) E L 1 ( -X,X)VX > 0 and

0.

a2 - 11 1Wz(b,a)l ~ C a2 - 11 +Jl(1 +

lbll----+ 0

as a----+ 0.

The final result follows by invoking Theorem 3.6.2 with iii( co) replaced by f/Jz(m).

An Application

3.8

We apply the preceding theory to the wavelet transform defined by means of the Mexican hat function l!'(x)

= (1 -

x 2 )e- ~x2 ; so that ij/( co) = (2n) ~ m2 e- !ro2 • Thus

W(b,a)

=

(2jn) 1 1 2 [~ eibro(am) 2 e-~(aro) 2 iii(m)dm.

(3.8.1)

Here (3.8.2) and

H(TJ)

= [~ (2n)~ ro 2 e-~ro 2 lml 1 - 17 dro = 2~( 5 -17ln~r(2- TJ/2), TJ < 4.

Therefore, by a modification of proof of Theorem 3.6.1, for TJ I

e-2ro

2~

(3.8.3)

< 4 and

I

t/)(m) E L (o,oo)\fo > 0, we have lim a2 -11W(b,a) a->=

and by Theorem 3.6.2, TJ lim a2 - 11 W(b,a) a--->0

= 2~( 3 -11ln-~r(2- !l)

lim iii(m)lml-l+17 2 lrol--->0

(3.8.4)

< 4 and m2 iii(m) E L 1( -X,X) VX > 0, we have

= 2~(3- 17 ln- ~ r(2- !]_) lim eibroiii( co) lmi-I+r!. 2 lrol_,=

(3.8.5)

Note that in the present case kernel ij/( co) is exponentially decreasing, hence conditions of validity of initial and final value results are relaxed. Results corresponding to those given in Theorems 3.7.4 and 3.7.5 can be obtained using results (3.8.4) and (3.8.3) respectively.

Chapter4


4.1

Introduction

In this chapter we consider the wavelet transform (3.1.2) for n = I and write

W(l/') = Wlf'(b,a) = _.!.._ { eibwij)(aw)ij(w)dw, 2n}R

(4.1.l)

where ij denotes the Fourier transform of function q,. The wavelet transform on Schwartz space Y'(IR.) was studied in Section 3.2. The spaces of typeS play an important role in the theory of linear partial differential equations as intermediate spaces between those of

c=

and of the analytic functions. The Fourier transform has been studied on the spaces of type S by Friedman [20] and Gel'fand and Shilov [21]. Nevertheless, there exist band-limited

wavelets with subexponential decay [15], and also infraexponential decay [76]; see Section 4.6. The aim of the present chapter is to study the wavelet transform (4.1.1) on these spaces. Let us recall the definitions of these spaces.

Definition 4.1.1. The space Sa(a

~

0) consists of all infinitely differentiable functions

lf'(x)( -oo < x < oo), satisfying the inequalities Yk,q( 1/') :=sup lx"q,(q) (x) I ~ CqAkJ! O,H

(2) Mp ~ RHPmino~q~pMqMp-q,

p E No,R > O,H > 0

(2a) Mp+l ~ RHPMp,

(3) Ej=oMj/MH 1
0

oo.

It can be easily verified that M P = pPa, a > 1, satisfies the above requirements. For each sequence { Mp} we define its associated function M(p) on (0, oo) by

M(p)

=

suplog(pPMo/Mp)·

(4.4.1)

p

If (Mol Mp) lfp is bounded below by a positive constant, M(p) is an increasing convex function in log p which vanishes for sufficiently small p and increases more rapidly than

log pP for any pas p tends to infinity. An example is M(p) = pa, 0 < a < I. Definition 4.4.1. The space of all those c=- functions f/J on JR. such that for every compact set K c JR., (4.4.2) for some constants C > 0 and A> 0, is denoted by G"(Mp;IR.).

Definition 4.4.2. The space of all which satisfy (4.4.2) is denoted by

c=

-functions on Q with compact support in K

c JR.

~(Mp;IR.).

Definition 4.4.3. Let {Mk}kENo and {Nq}qENo be the two sequences of positive numbers. An infinitely differentiable complex valued function f/J belongs to the space S{Mk} (JR.) if and only if k, q = 0, l, 2, ...

(4.4.3)

for some positive constants A and Cq depending on l/J; and l/J belongs to the space sl Nq} (JR.) if and only if k,q=O,I,2, ...

(4.4.4)


76

for some positive constants Band Ck depending on cp; and cp belongs to the space s~Z;j} (JR.) if and only if

k,q

= 0, 1,2,

0

0

0

(4.4.5)

where C,A and Bare certain positive constants depending on cp. Under certain conditions on {Mk} and {Nq}, the following relations hold: §[S{ Mk }]

= s{Md ,,fF[S{ Nq }] = s{Nq} • and §[S{Nq}] {Mk}

= S{Mk} {Nq}'

We shall need similar spaces of functions of two variables with certain restrictions on the order of differentiation. Definition 4.4.4. The space

s{M

I

p +qs

}

(H) is defined to be the space of all functions cp

E

C""(H) such that for alll,s,k,t,p,q E No,

1'/t.s.k,r(C/J) :=

laths(: )k (:b)t cp(b,a)l

sup

a

(h,a)EIR x!R+

~ Ck,rA~A~Mpt+qs•

t+s.;;k+t where the constants A1 ,Az and Ck,t depend on the testing function cp. Definition 4.4.5. The space

s{Nmk+m} (H)

is defined to be the space of all functions cp E

C""(H) such that for alll,s,k,t,m,n E No,

1'/t,s.k.r(C/J) :=

laths (

sup (b.a)E!Rx!R+

~

)k (

ua

~b)t cp(b,a)l ~ Ct,sB1B~Nmk+nt•

0

t+s.;;k+t where the constants 81 ,Bz and Ct,s depend on the testing function cp. Definition 4.4.6. The space S{{~mk+m }} {N pi +qs '

cp

E

gs+hl

} (H)

is defined to be the space of all functions

Coo(H) such that for alll,s,k,t,p,q,m,n E No, 1'/t.s,k,r(C/J) :=

1

la 1bs(: )k(:b) cp(b,a)l

sup (b,a)E!Rx!R+

l+s.;;k+t

ua

u

~ CA~A~B1 B~Mpl+qsNgs+htNmk+nt, where the constants A 1, Az, Bz, Bz and C depend on the testing function cp. Theorem 4.4.7. Let lfl E S{Mn}(IR.). Then the wavelet transform W is a continuous linear

map of s{Mn} (JR.) into s{M2k+,} (H). Proof. Let l,s,k,t E No such that 1+s

~

k+t. Then by inequality (4.2.1) we can write

laths (:ay (:by (wq,)(b,a)l


~

Using the property n!

nn and (4.4.3) we get

ia 1b.\' (:ar (:by

~

t

(~)sf

m=O X

X

n=O

~

n;2

f

(s ~

m) nn2t+k-/ 1m Cs-m-n (At+k-1-m-n lR

rDm+l-p-q(A~+k-p-q

(m)pp2m mEp (m+l- p)qq2k [ P q=O q }IR

m+k-p-q+2 ) dy ] Mm+k-p-2 +AI Mm+k-p-q-2 I+ y2

t (s)

m=O m X

(w~)(b,a)l

dw M t+k-1-m-n +A t+k-/-m-n+2Mt+k-l-m-n+2 ) I + w2

p=O X

77

m~- P i..

q=O

E

s f (s-m)nn2t+k-Lc;_m-nAt+k+2Mt+k+2 (m)pp2m n=O n p=O P

(m + l - P) qq2kDm+l-p-qA!m+k+2 Mm+k+2 1

q

~ n:2ss2t+k-Lc:~'At+k+2Mt+k+2 m~O (~) :~ e~m)mm2m(m+l)m+t

x2kDm Am+k+2Mm+k+2 -f.. (m) m~- (m + [ - p) s.l

J..

I

p=O

p

J..

P

q=O

q

·

Now using inequalities (Ia) and (2) the last expression can be shown to be

~ Es,iB1B~Mt+2k> where Es.t,BtandB2 are positive constants. The following two theorems are generalizations of Theorem 4.2.2 and Theorem 4.2.3 respectively. Their proofs are similar and exploit the technique used in the proof of Theorem 4.4.7, hence are omitted.

Theorem 4.4.8. Let 11' E

s{Nm} (JR). Then the wavelet transform W is a continuous linear

map of S{Nm} (JR) into S{Nt+ 2,} (H).

78


Theorem 4.4.9. Let 1f1 E S~~~~ (IR). Then the wavelet transform W is a continuous linear map oifs {Mn} {Nm}

(lTJ)) . m.

S-{N2k+t}

mto {Mt+2s},{J\Ir}

(H)

.

Remark 4.4.10. Using the technique employed in Section 4.3 we can define the wavelet transforms of ultradistributions belonging to (S{Mp} )', (s{Nq} )'and (s~Z~} )'.

4.5

Wavelet Transform of Gevrey Functions of Compact Support

In this section we shall investigate the wavelet transform of Gevrey functions of compact support denoted by Gb(.Q), where .Q is an open subset of JR. First, we recall the definition of this space as given in Rodino [84]. Definition 4.5.1. The function f(x) is in cs(n.) if f(x) E C"(.Q) and for every compact subset K of .Q there exists a positive constant C such that for all p E No and x E K,

lf(Pl(x)l ~ CP+I (p!)". Definition 4.5.2. Assume s > I. We shall denote by Gb(.Q) the vector space of all o~(.Q)

f

E

with compact support in .Q.

We shall assume that the Fourier transform of the wavelet 1f1 satisfies certain general growth conditions and it belongs to the space s;;:;(JR) defined below. Definition 4.5.3. We shall denote by

s;:; .(1R) 1

the space of all functions f(~) E Coo(IR)

satisfying the following condition: for every compact subset K

c lR there exist constants

C, B > 0, and for every £ > 0 there exists a constant Ce such that IDU(~)I ~ C:Cq(q!) 1 (l

for every q E No

and~ E

lR with

I+~~

+ 1~1)-Pqexp[£1~1 1 /s],

Bqs, where s,t,p are real numbers such that s >I,

O~p~l.

Definition 4.5.4. We shall denote by Bs(H) the space of all complex valued infinitely differentiable functions

Ibm (:a

q, (b, a) that satisfy the following condition :

Y(:bY

cf>(b,a)i

~ CeAmBkct (m!)P(k!)q(l!Y(l +a)mexp[ea fs], 1

whereA,B,C,Ce are positive constants, p,q,r ~ 0, I +k+ I> m and£ is any small positive number. Theorem 4.5.5. Jffji

E

from Gb(IR) into Bs(H).

s;;'}s(IR), then the wavelet transform W is a continuous linear map

79


Proof. Proceeding as in the proof of Theorem 4.2.1, we have I= bm (:ar (:by

(w~)(b,a)

r .b ~ ( a (a )1liRe'

= bm aa )k db = ;m+t X

=

k

eibw

P~ (;) { (:a) k ( :w) PV/(aw)}

q~~-p (m; p) { ( :W

im+l

r eibw [,

}JR.

-~

w~(w)lfl(aw)dw

r 1}(:W) w

m-p-q ~(w)dw

(m){tr=O (k)r (p ~! r).,ap-rwk-rV/(p+k-r)(aw)}

p~m p

xq~~-p (m; p) {(I ~!q)! w -q} ( :w) m-p-q ~(w)dw. III ~ r [, (m) {t (k) ~! ,ap-rlwk-rVI(p+k-r)(aw)l} JJR.p~m p r=O r (p r). 1

x

~

q~~-p (m;p) { (l~!q)! wt-q }1 (:w )m-p-q ~(w)ldw

r [, (m) t (k)

p! ,ap-rwk-rcecp+k-r((p+k- r)!/ JIRp~m p r=O r (p-r). X

(l+lawl)-p(p+k-r)exp[t:lawll/2s]

x w1-qCm-p-q.hexp[-hlwl 1l']dw,

~

h > 0 [84,p.58]

[, (m) r=OE(k)r (p - r) ·,ap-rcecp+k-r((p+k- r)!)t p!

p~m

P

X

wk-r(l + lawl)-p(p+k-r)exp[t:lawl 1/ 2s]Cm.p.h

X

~

[, (m-p)_l_!(l - q) ! q q~m- P

k

[, q~m-p

(m-p)q!iw 1-qexp[-hlwl 1fs]dw q

[, (m) r=OE(k)r!2PCeCp+k-r((p+k-r)!Yap-r r

p~m

P

X

lwlk-p(I

k

+ lawl)-p(p+k-r) exp[t:lawl 1/ 2s]Cm.p.h(m- p)!

x 21w1exp[ -hlwl 1fs]2m-p dw

~ Cecm+kcm,hJl+m t

p~m

(m) r=OE(:)r!((p+k-r)!Y(m-p)!ap-r P

80


X

k

lwll+k-p(1

+ lawl)-p(p+k-r) exp[elawl 1/ 2s- hlwll/s]dw

~ CeCm,hcm+ki+m f. X

k

p=O

lwll+k-p ( 1 + lawl)-p(p+k-r) exp[elawl 1/ 2s- hlwll/s]dw

~ CeCm,hcm+ki+m f.

p=O

X

t (~)r!((p+k)!) 1m!ap-r

(m) p r=O

t (~)

(m) p r=O

r!((p+k)!) 1m!ap-r

k_lwll+k-Pexp[~(lall/s + lwllfs) -hlwllfs]dw.

for all h > 0 (see [84], p.58). Choosing e = h, we can estimate this expression by ChCmcm+ki+mk!m!((m+k)!l

f.

p=O

where D

= fJR lwll+k-Pexp[-~lwl 1 1s]dw.

t (~)ap-re~ 011'D,

(m) p r=O

Thus

III ~ DChCm(C21+2 )m(C21+l / i (k!y+t (m!y+t (1 + a)me~ 011'. 4.6

Band Limited Wavelets

This section contains examples of wavelets belonging to certain Gevrey classes. The following generalized version of the Paley-Wiener theorem [38, p. 137] will be used in the sequel.

Theorem 4.6.1. Let 0 > 0. Suppose thatfis afunction such that suppf C [-A,A], A> 0, and f E G 5 . Then for every B > 0 there exists a constant CB such that

Definition 4.6.2. A real valued function defined on

~

is said to have

(i) exponential decay if there exist C > 0 and A > 0 such that VxE~;

(ii) subexponential decay if for 0 < e < I, there exists Ce > 0 such that VxE~;

(iii) infraexponential decay if there exist C and A > 0 such that VxE~.

81


Following the technique ofDziuban'ski and Hema'ndez [15] we construct an orthonormal wavelet in L 2 (JR) with sub-exponential decay and whose Fourier transform has bounded support; such wavelets are called band limited. Spline wavelets have exponential decay. But there is no orthonormal wavelet with exponential decay belonging to C"'(!R) such that all its derivatives are bounded. Now, with X= X[O,l] set Xa

= (1/a)x(x/a).

Then for any sequence a1 ;.;,: a2;.;,: a3 · · · > 0

I:}= 1 aJ < oo, the function

such that a =

cf>k

=

Xa 1 * Xa2 · · · * Xak

belongs to ck-l (JR), has support in [O,a] and converges as k ----+ with support in [O,a], such that fiR cf>(x)dx = I and 2n

IDncp(x)l

= n- 0

By taking an

it follows that

q,

~

oo

to a function

q, E C"' (JR.),

·

aJ ... an E C0 when D > I.

For fixed a> 0, choose a "cut off' function cf>a E C0 for every D > I, where cf>a(x) =

(lja)cf>(xja). Set Ba(x)

= J~oocf>a(t)dt.

Then Ba E C 0 for every D > I. Now, let Sa(x)

= sin(Ba(x)) and Ca(x) = cos(Ba(x)),

and define the "bell" function associated with the interval [tr, 2tr] by

ba(x) = Sa(x -tr)C2a(X- 2tr),

0 I. Extending ba evenly to [-oo, OJ it follows that the function

li'a defined by

is an orthonormal wavelet in L 2 (JR). Moreover, by Theorem 4.6.1,

lll'a(x)l = Clba(x+ 1/2)1 ~ Ceexp( -lx+ 1/21 11°) ~ Ceexp( -lxl 1-e), x E JR., where 0
(z)l

~ Ckexp(blyl)'f(l-f:l)_

This coincides with the space sf:l. Example3. LetM(x) =x 1fa,n.(y) =y 1f(l-f:l) (0

(x+ iy)l This coincides with the space

~ Cexp[-alxl 1fa

+ blyi 1/(I-f:l)].

sg.

Theorem 5.2.4. If the functions M(x) and .Q.(y) are mutually dual in the sense of Young, then the Fourier operator~: WM,a--+ wn,~ .~: wn,{:l--+ wM,k iscontinuousandWM.a = WQ,~, Wil,f:l = WM 1 • 'lf

Theorem 5.2.5. Let .Q.I (y) and M1 (x) be the functions which are dual in the sense of Young to the functions M(x) and .Q.(y) respectively, then the Fourier operator~: wf:! --+ Wn 1 '

'f

MI·7J

. contmuous . ts an d w:~n M'af:l = wn"~1 • .

Ml,lf

In what follows we shall also need the following similar test function spaces, called spaces of type Wwhich will be used in the study of the continuous wavelet transform. Definition 5.2.6. The space WM,a is defined to be the set of all complex valued infinitely differentiable functions cf>a( CJ)

= cp( CJ,a) E Coo (JR. X JR.+)

which for any 0

> 0 satisfy

l(cfa)k(ja) 1cp(CJ,a)l ~Ck 18 e-M[I~a(a-o)]; k,l=0,1,2, ... ,

(5.2.8)

where positive constants Ckto depend on function cp. Definition 5.2.7. The space W*n,n,{:l,af:l is defined to be the set of all functions cf>a(s) =

cp(s,a), (s,a) E C x JR.+, entire analytic with respect to s

= CJ+ ir which for any p,p' > 0,

satisfy

l( _§_)k"'(s I +a 'I' '

a)l ~Ckpp' en[-r(f:l+p)]+n[-r(af:l+p'J].• k=O , I , 2 , · · ·, '-"

(5.2.9)

where positive constants Ckpp' depend on function cp. Definition 5.2.8. The space wn,n,{:l,f:l is defined to be the set of all functions cp(s,t), (s, t) E C x C, entire analytic with respect to s = CJ + ir, t

= a+ iy which for any PI, pz > 0 satisfy

I(;fs)k(;~dct>(s,t) I ~ CktPIP2en[-r(f:l+Ptl]+il[r(f:l+P2ll; where positive constants Cktp 1p2 depend on function cp.

k, l

= 0, 1,2, ... '

(5.2.10)

86


5.3 The wavelet transformation on W -spaces In this section we study the wavelet transform (4.1.1) on the spaces WM,a·

wn,f3, and w:f.'~.

In what follows, for convenience, we impose conditions on (j), but one can impose conditions on

f/J, and apply appropriate Theorem 5.2.4 or 5.2.5, and get the desired result.

Theorem 5.3.1. Let M(x) and Q(y) be the functions which are dual in the Young sense. Suppose fi/ E WM,a and (j) E WM,a. then the wavelet transform <Pa(s) I

I

w• 0 ' 0 'a"aa, s =

(j

+ ir,

= cP(s,a)

E

for arbitrary but fixed a > 0, that is, wavelet transform -nnl

__L

('W'I'f/J) (s, a) is a continuous linear map from W 0 •I I a into w· , ,a ,aa

0

Proof. The wavelet transform of a function f/J with respect to the wavelet If/ defined by (4.1.1) is

('W'I'f/J)(cr,a) =cP(cr,a) = -21 7r

Joo eiax{j;(x)fi/(ax)dx.

(5.3.1)

-00

Since (j), filE WM,a,therefore, the wavelet transform can be extended to the complex values of s = cr + ir according to the definition

cP(cr+ir,a) =

_!_joo ei(a+i'r)x{j;(x)fi/(ax)dx 27r _21 !oo eisx{j)(x)fi/(ax)dx. 7r -00

=

-00

Integrating by parts k times, we get

[(is)kcP(s,a)[

=I (~;k l~ eisxD~({j;(x)fi/(ax))dxl = -1 t (k) 100 )k

I(

27r

=

~(-1)k 27r

~

eisx oik-1) {j;(x)DY) fi/(ax)dxl

1=0

l

-00

1=0

l

-00

t (k);·oo t (k) joo

eisx(j)(k-l)(x)alfjl(t)(ax)dxl

_!_(1 +a)k 27r

Now, using Definition 5.2.1, we get

1=0

l

-00

[i1x(j)(k-l)(x)fi!Ul(ax)[dx.

(5.3.2)

87


C , (k-1)8 18 l f::o (k)c

< ~f. "' 27r

1=

< C' "' ~~ .

1=

e2]n]-M[x(a-8)]-M[ax(a-8')ldx

-=

e2lni-M[x(a-8)]-M[ax(a-8')Jdx

-=

'

q

where 88 , = 2~ E7=o (7)c(k-l)8Ct8'· Now, applying (5.2.3) and Young's inequality (5.2.4), the exponent in the above integral can be transformed as follows:

-M[x(a-8)]+1rxl ~ -M[x(a-8)]+M[x(a-28)]+D. [a_'1"28 ]

~ -M[8x]+n[a_'[28 ], and

~ -M[ax(a- 8')] +M[ax(a- 28')] +D. [ a(a~ 28 ,)]

-M[ax(a- 8')] + lrxl

~

-M[8'ax] +D.

[a(a~ 28 ,)].

Therefore, we get the estimate

III ~ c~88'en[a_'[28]+n[a(a~28')] < C" "'

kpp'e

In the above we set a~ 28

=

I:

n[T(!k+P)]+n['r(a~+P')]

!k + p and

a(a~ 28 ,)

e-M[8x]-M[8'ax]dx

.

= a~+ p' where the quantities p,p' are I

I

arbitrary small together with 8, 8'. Thus cP(s,a) E W*.n,n.a'aa.

Theorem 5.3.2. Let the functions M(x),D.(y) be same as in Theorem 5.3.1. Suppose ljJ E wn.,B and iii E wn,,B. Then wavelet transform ('Wifllf> )( cr,a) is a continuous linear map from wM.I/,6 into WM,I/,6• that is, cP(cr,a) E WM.lj,B·

Proof. Since iii E wn,,B, following the technique of [22, p. 22], the expression for wavelet transform defined by (5.3.1) can be written as cP(cr,a) =

~1= eiu(x+iy)iii(x+iy)lj/(a(x+iy))dx 27r

= ~1= 27r

eiuziii(z)lj/(az)dx,

z=x+iy.

-=

For non-negative integers k, /,after differentiation of (5.3.4 ), we get

C/u/(J) 1cP(cr,a)

(5.3.3)

-=

=

2~ I: eiuz(idiii(z)(jjljl(az)dx

(5.3.4)

88


= !___ joo eiazl+Lif(z)iji(L)(az)dx. 27r

Now, using inequality lzl 1 ~

-00

izi 1x+ ~2 +1izi 1 and conditions for belonging lj, ijf in wn,/3, we obtain

III= lc/a/(fY~(a,a)l ~ 2~

L:

leiazl ('zik+:2:;izik+L) 1/F(z) II ij!U) (az) Idx

joo e-ay[Ck+L+2 p +Ck+L p]en[y(f3+Pllco 'Len[ay(f3+p')]~ x +1 :>:: C' e-ay+.O[y(J3+p)]+.O[ay(f3+p')] foo ~ ""' klpp' x2 + I :>:: _.!._ ""' 27r

-oo

'

P

•

2

-oo

~ ~lpp'e-ay+.O[y(f3+p+a(f3+p'))] ~

c:lpp'e-ay+.O[y(l+a)(f3+p)]

for all p, p'

>0

for p = p'.

(5.3.5)

Until now y has been an arbitrary number. Using the technique [22, p. 22], let us now choose the sign of yin such a manner that the equality ay = IallYl be satisfied, and absolute value of y so that the Young inequality (5.2.4) becomes equality laiiYI

= .Q[Iyl(l +a)(J3 +P)] +M [(t+a1J(1+Pl] ·

Then exponent in the expression (5.3.5) becomes

+ a)(J3 + p )] =

-cry+ .Q[y(I Replacing 1I (J3

+ p) by ( 1I J3) -

-M [ (l+a1)(1+p) J .

o where o is arbitrary small, we obtain the estimate for

the expression (5.3.5):

I( daa )k(l__)L~(a au ' a)l :>::C" -.. : klo e-M[(i~~)(~-o)] . Hence, wavelet transform ~(a,a) E WM,I//3·

Theorem 5.3.3. Let the functions M(x) and .Q(y) be same as in Theorem 5.3.1. Suppose l

ijf E wn·lJ and If E WM.f3·Then wavelet transform (ll''l'cp)(s,t) extends to an entire function . =a+ q. an d.lt zs. a contznuous . l.znear map firom wnJ . w~n,n.i.~ oif s =a+ rr,t ~-' mto ~-' ~-' Proof. The wavelet transform of a function cp with respect to the wavelet 1JI defined by (5.3.1) is

~(a,a)

= -21

7r

joo eiax/F(x)ij/(ax)dx -00

89


Since both {j and 1f1 E WM,/3• from [22, p. 20] we have

(cr+i-r,a+iy)

=

__!__1oo ei(a+ir)x{j(x)1oo ei~(a+iy)xlJI(/;)d/;dx; 27r

-oo

-oo

so that

r

__!__ 100 eisx{p(x) ei~txlfl( I;) dl; dx. 27r -oo .J-oo Fork, l E No, after differentiation of (5.3.6), we get

(s,t)

=

(ft )k ( -Js )l(s, t) = 2~ =

i:

ei'x(i4 ;p(x)

i: ei~tx(ixl;

(5.3.6)

)k lJI( I;) dl; dx

__!__ 1oo eisx(ix)l+k{p (x) 1oo ei~rx l;k lfl( I;) d/; dx. 27r -oo -oo

Now using the definitions for (j and lJI, we obtain III= ,;;

,;;

j(frl(-#s) 1(s,t)J

i:

i:

2~ leisxll (ix)l+k{p (x) I lei~txlll;k lJI( I;) Id/; dx 2~ j_: e-•xcokt8e-M[x(f3- 8)] j_: e-rx~Cok8'e-M[~(f3- 8')J dl; dx

~ 17r COk/8 COk8 1oo elrx]-M[x(f3-8)]1oo elrx~]-M[~(/3-8')1 d'" dx -oo -oo o., • 2 1

--=:

(53 7) • •

Applying Young's inequality (5.2.4) in the of exponent (5.3.7),we get lnl- M[x(J3- 8)] ,;; Q[/3~~ 8 ] +M[Ixi(J3- 28)]-M[x(/3- 8)]

,;; n[ 13 _:28 ]-M[ox] and

lrxl; 1- M[l; (J3- o')] ,;; Q[/3~~(5,] + M[lxl; I(J3- 28')]- M[l; (J3- o')]

,;; Q[ /3__1; 8,]- M[/; ((J3- o')- x(/3 - 28') )]. Setting ~ =

b

b

+ p and ~ = + p' where p and p' are arbitrary small together with 0 and 0 1 , inequality (5.3.7) becomes III ,;; __!__Cokt8Cok8'en[{3_:zo] en[/3-~ 8 ']1 00 e-M[ 8xl1oo e-M[~((/3- 8 ')-x(/3- 28 '))] dl; dx 27r -oo -oo

~ C'ktpp'e n[r(b+P )] en[r(b+P')]I 1, "" where I1 denotes the double integral which is estimated as follows:

Il =

i:

e-M[8x]

i:

e-M[~((/3-8')-x(/3-28'))] d/; dx;

(5.3.8)

90


-1oo -M[ox] 1oo e-M[u] d d -oo e -oo (/3 - 8') - (/3 - 28')x u x -1oo -M[u] 1oo e-M[ox] -oo e du -oo (/3 - 8') - (/3 - 28')x dx =

-A 100 e-M[ox] f3 _ 28 , -oo x- (J dx; wherelJ

f3

=~ 8 ,H(e-M[oxl)(lJ);

o'

=

~~20 ,,A

100 =

-oo e-M[uldu

where H denotes the Hilbert transform.

Since e-M[ox] E LP(IR) for p > 1, from [96, p.275], we have

Since the same LP estimate is valid for first order derivatives, from the Sobolev embedding theorem we have that Hilbert transform H (e-M[ox]) ( (J) belongs to Loo. Hence

IIII

~B(f3,8,8',p).

-

-nn.!..!.

Consequently, it follows from (5.3.8) that (s,t) E W ' 'f3 'f3. Theorem 5.3.4. Let .Ql (y) and M1 (x) be the functions which are dual in the sense of Young

to the functions M(x) and .Q(y), respectively. Suppose 11' E WM 1 ,f3 and -

fixed a E IR+ the wavelet transform (s,a), as a function of s = where A = f3

if E w1J:~- Then for a I

Q

0"

+ i't", belongs toWM1'f, I•I

+ ~-

Proof. By [22, p.24],the expression for the wavelet transform of the function

if E w1J:~ can

be written as

( 0" + i't", a) =

__!___

2n

ei~a(x+iy) 11'( ~) d~ dx;

1oo ei( 0.

(2) For

iii E Wq,f3 and iii E Wq,f3 (or 1/f E Wp,l//3 and l/J E Wp.Ijf3), 1f/"'[wp,1Jf3l

(3) For

= wp.ljf3·

iii E Wq,l//3 and iii E Wq.f3 (or 1/f E Wp,f3 and l/J E Wq,l/f3), 1f/'I'[Wq,lff3]

(4) For 1/f E Wr./3 and

iii E WJ,'g

= Wq,q.(l//3),(1//3).

(or 1/f E Wr,/3 and l/J E W:]Jf3a),

11'; [Ws,lja] = Ws,lja A.= 'If

r,l//3

r,ljA. '

f.!

1-'

+aiR. 1-'

Chapter6


6.1

Introduction

Let II' E L2 (1R.) be an analyzing wavelet which satisfies (1.6.3). We define the translation operator Tb by

Tbl/f(x)=l/f(x-b), hEIR.,

(6.1.1)

= lal- 1/ 2 11' (~),

(6.1.2)

the dilation operator Da by

Da 1/f(x)

a E IR.o

=JR.\ {0}

and a unitary transformation U(b,a): L2 (1R.,dt)----> L2 (1R.,dt) by

U(b,a)l/f(x)

= (TbDal/f)(x) = lal- 1/ 2 11' ( x:b),

(b,a) E JR. x IR.o.

(6.1.3)

The actions of the Fourier transform on the operators Tb and Da are given by (6.1.4) (6.1.5) Then the integral

(Lv,J)(b,a) =

1

frW'I'f(b,a) =

(6.1.6)

yC'I'

yC"'

1 = fr

1

fr(f,U(b,a)l/f)

1 1m JT::T

yC'I' y

ial

f(t)l!'((t- b)/a)dt

(6.1.7)

IR

defines an element of L 2 (JR. x IR.o, d~~a) . Moreover, L"': L2 (1R.,dt)----> L2

(JR. x JR.0 , d~qa) 93

is an isometry [45, Theorem 2.2, p. 30].

94


The Fourier transform of L"' with respect to its translation argument is given by

(L"'f)(·,a/'(~) =

;,___

yC"'

alal-l/Zo/(a~)f(~).

(6.1.8)

The operator L"' is also called wavelet transform with respect to analyzing wavelet II'· This is a normalized form of the operator W"', considered earlier. In this chapter, we extend the wavelet transform L"'(f), which we defined on L2 (JR,dt), to generalized Sobolev space B~k and interpret its image set as the space U((lRo, !f!i),B~k) abbreviated by Wp~k· This work generalizes some of the results contained in [81].

6.2 The generalized Sobolev space B~k In this section we recall definitions and properties of certain function and distribution spaces introduced by Bjorck [5]. Let Jlt be the set of continuous and real valued functions w on JRn satisfying the following conditions:

(I) (6.2.1) (2)

r (I+~~ w(~) l)n-t-1 d~ < oo,

}JW.n

(6.2.2)

(3) (6.2.3) for some real number a and positive real number b. We denote by Jltc the set of all w E Jlt satisfying condition w( ~) = Q( I~ I) with a concave function Q on [0, oo). We suppose

w E Jltc from now on. We denote by Sro the set of all functions lfJ E L 1(JRn) with the property that lfJ and and for each multi index

($ E C"'

a and each non-negative number A we have Pa.A.(l/J) =sup eA.ro(x)IDalfJ(x)l < oo;

(6.2.4)

xEIR"

1ra.A.(lfJ) = sup eA.ro(~)IDa($(~)1 < oo.

(6.2.5)

~EJRn

The topology of Sro is defined by the semi-norms Pa,A. and 7ra,A.· The dual of Sro is denoted by S~, the elements of which are called ultra-distributions. We may refer to [5] for its


various properties. We note that for ro(;) = log(1

95

+ 1;1), Sro

reduces to!/, the Schwartz

space. We also recall the definition of test function space D 00 • The space Dro is the set of alief> in L 1(!Rn) such that cp has compact support and lll/>IIA.

llct>IIA. =

< oo for all A> 0 and

f l~(;)leA.ro(~ld;.

}JRn

(6.2.6)

Now, let ro E .4c. Then Kro is defined to be the set of positive functions k in IRn with the following property. There exists A > 0 such that (6.2.7)

< oo. Then generalized Sobolev space B~.k(IRn) is defined to be the space of all ultra-distributions f E S~ such that

Let roE .4c. k E Kro and 1 ~ p

(6.2.8) and 11/lloo.k = esssupk(;)IJ(;)I.

(6.2.9)

Note that the space B~,k(IRn) is a generalization of the Hormander space Bp.k(IRn) [38] and reduces to the space Bp.k (IRn) for ro = log( 1 + I; I). For k(;) = esro( ~) and I ~ p ~ oo;

the generalized Sobolev space studied by Pathak [63]. An inner product in B~k is given by (6.2.10) The space B2.k(1Rn) reduces to Sobolev space Hs(IRn) fork(;)= (1

+ 1; I2 )S1 2 • The Sobolev

space Hs(IRn) consists off E Y'(!Rn) such that (6.2.11) where s E JR.

6.2.1

Examples

1. Dirac

o belongs to W(IRn) for s < -n/2.

2. Let g(x) = Eaaao(a)(x- a), aa E C. Suppose that, for each e > 0, there exists Ce. such that laal ~ Ceelal(a!)-lfd, 0

< d 0 and suitable constant De. Therefore, g(x) E B~k(IRn),

wherero(~)=l~ld,

But g(x)

O s, where s E IR, is fixed. Consider the convolution of

3. Let mE No, such that m+ 1/2

m + 1 characteristic functions:

g := X[o.t] Then, from [2, p.487],

* ··· * X[o.I]·

g(~) = e-i(m+l)~/ 2 cinJ% 2) r+l, and g E H'.

4. Let g be the same as in Example 3. For every j = 0, 1, 2, ... ,define the wavelet q,UJ by

(i)Ul (~ ) =

g( ~ ) Jrog,J(~)

with 00

ro8 .j(~) = [. (1 +2 2 1(~ +2kn) 2 Yii(~ +2kn)l 2 . k=-oo

Since there are constants

c1,c1 > 0 such that c1 < Wg,J < c1, it follows

that q,Ul(~) E

Hs(IR), cf.[2, p. 488].

6.3 The wavelet transform on generalized Sobolev space B:,k In this section we define the space llf(b,a)llwf = 1~ p

w;: of all measurable functions f

d ) (liRor (llf(b,a)ll:,k) laiP;+l

ljp

on IR x !Ro such that

< oo,

(6.3.1)

< oo, a E IR\ {0} = !Ro.

Theorem 6.3.1. Assume that analyzing wavelet satisfies the following admissibility condition:

c

= lfl.p

r lo/(~W dj: < I

lJR

I~

~

00

(6.3.2)

.

Let (Liflf)(b,a) be the wavelet transform of the function f E

B~.k

with respect to the ana-

lyzing wavelet If! satisfying (6.3.2). Then (6.3.3)

97


where Ap = (C'I')-PI 2 clfl,p is a positive constant depending upon p and 'I'· Proof. Assume at first that f E S00 (1R.). Then from (6.3.1), we have

ko II(L'I'f)(b,a)ll~,k lal(:~)+l ko (k lk(~)IPI(L'I'f)(b,a)i\(~)IPd~ lal(:~)+l) 2 = ko (k lk(~)IP (~'I') pf laiP! 2 Io/(a~)IPIJ(~)IPd~) lai(:/~)+I

II(L'I'f)(b,a)ll~: = =

1 = (-

c"'

= (

)p12 liR.or lo/(u)IP dull/liP lui

p.k

~"') p/ c"',pll!ll~,k = Apll!ll~,k· 2

Since by [5, Theorem 2.2.3, p. 384] Sw is dense in

B~k·

the above result can be extended

to all f E B~,k·

6.3.1

Adjoint of L"'

Let Wk and Bk denote the spaces Wl and sr,k respectively which are the special cases of the spaces W{ and B~k for p = 2. If k( ~)

=I, then Wk is denoted by W1. Assume that 'I' is

real valued. From (6.3.3) it follows that the operator L"' with integrable and admissible 'I' and p

= 2,

is a linear isometry from Bk to Wk. For linear isometry between Hilbert spaces we know that U * U = id and UU* is the orthogonal projection onto range(U) (which is closed), where U* denotes adjoint of U. Therefore, the transform L"' is inverted onto its range, by its adjoint L; and that an element g E Wk lies in range(L"') if and only if L"'L; =g. Suppose that

L; is the adjoint of transform L"'. Now, we derive an explicit expression for L~: Wk----> Bk.

We assume that f E S00 , and set (6.3.4)

and (6.3.5) Setting up a scalar product on Wk:

(Q, r)k =

1m

IRo

da (Q(·,a), r(·,a)}k--z a

(6.3.6)

98


we get

(Ly,f,g)k= f f 1\(a,~)d~d~· 1JR0 JJR a

(6.3.7)

Applying two times Schwarz's inequality, we get

r JJRrIA(a,~)ld~d~a ~ 1JRr0 IILI{/f(·,a)llkil(g(·,a))ll/~a liRo ~ IIL"'/IIwk llgllwk'

(6.3.8)

which allows the change in the order of integration in (6.3.7). We have from (6.3.5) and (6.3.7),

(Liflf,g)k = { (k(~)) 2 j(~)d~ { ~C (D-al/f)'\~)(g(·,a))A(~)d~· lJR 1JR0 y '"'If/ a

(6.3.9)

We denote the inner integral by Ag( ~) and estimate lAg I to conclude that Ag E L2 (1~, dt). Now,

-

Ag(~)

= 1m

I A A da IC(D-al/f) (~)(g(·,a)) (~)2.

~Y'"'Ifl

(6.3.10)

a

Using Schwarz's inequality in (6.3.10), we have

IAg(~)l 2 ~ 1JRf l(g(·,a))t\(~Wd~· a

(6.3.11)

0

Therefore,

f IAg(~)l 2 d~ ~ f f l(g(·,a))t\(~)l 2 d~ d~

JJR

lJR 1JR0

a

=

llg(·,a)ll~1 •

Consequently, there exists an AgE L2 (l~,dt) with (6.3.12) Using (6.3.12) in (6.3.9), we have

(Liflf,g)k = =

k(k(~)) j(~)(Ag)A(~)d~ 2

(f,Ag)k.

(6.3.13)

Thus Llfl and A are adjoints of each other on Bk. Next we determine an explicit form of Ag(x) using the fact that the integral

fJR.o fJR(D-ali')A(~)(g(·,a))t\(~)d~ '1ft exists. We have

99


Therefore

Ag(x)

6.3.2

=

B~,k -approximation

;1r 1m y Cll'

1m

IR0 IR y

1 lfl (b-x) dbda JT::T g(b,a)2 .

Ia I

a

(6.3.14)

a

of wavelet transforms

We will now determine the B:,k -distance of two wavelet transforms with different basic wavelets and different argument functions in order to study the dependence of the transform on its wavelets and its argument. 00 k, Theorem 6.3.2. For admissible and integrable lf/1, lf/2 and f, g E B p,

IILII'J(b,a) -LII'2 g(b,a)llp,k

"lal 112

(II }c;-

;.;t

IIIII,,.+

I

;.;t

11/-gll,>) ·

Proof. We have

IILytJ(b,a) -Lyt2 g(b,a)llp,k ::s;IILII'1f(b,a) -LII'zf(b,a)llp,k

+ IILyt2 /(b,a) -Lyt2 g(b,a)llp.k·

(6.3.15)

Now,

IILII'J(b,a) -LII'zf(b,a)llp,k

=

(k

I(LytJ(b,a)

-Lytzf(b,a))"'(~)IPik(~)IPd~) l/p

~ (J.I )c..1al 112 o/i(a~)j( ~)- ;;;-lal 112 ¥ii(a~ lfC ~)I' lk( ~)I'd~)

~ (J.wt'lk( ~)I'll@ I'd~ I( Jb- Jb) (a~f) 11, Now, using inequality

I/P

(6.3.161

100


we have

so that (6.3.17)

Using (6.3.17) in (6.3.16), we have

IIL.,f(b,a) -L.,f(b,a)ll,> 0.

Let us write lf!a(x) =

~ 1{1 G),

(6.4.2)


101

and use the notation:

1\y,f(b,a)

r (b-t) = (lfla *f)(b) =;• J!Rf(t)lfl -a- dt.

(6.4.3)

Notice that in previous chapters 1\IJf was also denoted by Wx, where x(x) = lfl( -x). From (6.4.1) and (6.4.3), we have

(lfla*f)(b) = (1\IJff)(b,a) =

yrc; -;-LIJff(b,-a).

(6.4.4)

Let P(D) = E'J=oCjDj be a differential operator, where Cj E C and Pis polynomial. Define

(P(c;)f =

m

E IDjP(c;)l

(6.4.5)

2.

j=O Then, by Hormander [38, p. 10], we have (i) f E B~k ==? P(D)f E B:k/P

(ii) IIP(D)fllp,k/P:::;;

11/llp,k/P"

Lemma 6.4.1. Let f E B~,k and lfl E L 1 (~)with fiR lfl(t)dt

= 1. Then

(i) 1\IJff(·,a)---. f(-) in B~,k as a---. 0; (ii)

(6.4.6) Proof. In view of (6.4.3) and (6.1.8), we have (i)

lllfla * f- fll~,k = ~ l(lfla * f- f)'\c;)IPik(c;)IPdc; =

~ l(lfla *f) 11 (c;) -J(c;)IPik(c;)IPdc;

=

(c;) ~ I( C: )

=

~ lif/(ac;)J(c;) -J(c;)IPik(c;)IPdc;

=~I

1/2

((LIJff)(b,a)) 11 (c;) -J(c;) 1/2 (

diJf

) 1/2

lp

lk(c;)IPdc;

lp

lal 1/ 2 if/(ac;)J(c;) -J(c;) lk(c;)IPdc;

= ~ lk(c;)IPIJ(c;)IPil- i/f(ac;)IPdc;

= ~ lk(c;)j(c;)IPII- i{l(ac;)IPdc;

The Wavelet Transfonn

102

=

kII(a,~)IPd~,

where II(a,~)l = lk(~)j(~)(1- if/(a~))l. Then we have lima ..... o II( a, ~)I= 0 a.e. Let us now set M = sup~EIR 11

-

i/f(a~) I, which is independent of a.

Then

Now, applying the dominated convergence theorem, we have

(lfla*f)=l\lflf(·,a)--"f(·) in B~,k as a--'>0. (ii) Let {fn}nEN E S00 (1R.) converge to fin B~.k· Now, operating the differential operator P(D) on both sides of equation (6.4.3), we have the following equality.

Since S00 (1R.) in dense in

B~k(IR.)

and the operators 1\IJI and P(D) are continuous, hence

limits of the above three are equal in B~,k· Theorem 6.4.2. Let f E B~,k and lfl E B~.k nL 1 (JR.) with fiR lfl(t)dt = 1 and P(D)lfl E L 1 (JR.).

Then lim

a-->0

where cd

=

1 a 112 LP(a-'D)lflf(b,a)- v~P(D)/11 _ 0, cd =

1

p,k/ p

CP(D)IJI•

Proof. By virtue of Lemma 6.4.1 (ii), we have

LP(a-'D)IJif(b,a)

Now, we estimate

=

~ 1\P(a-lD)IJif(b,a)

=

~ 1\IJIP(D)f(b,a)

=

~P(D) 1\lfl f(b,a).


103

=II a1~2 ~P(D)(Aytf)(b,a)- ~P(D)ftk/P I

=Jed IIP(D)[Aytf(b,a)- f]llp.k/P I

~Jed II Aytf(b,a)- f(b)JIIp.k.

by (6.4.6), and according to Lemma 6.4.I (i),

Aytf(·,a)---+ f(·) as a---+ 0. Therefore,

Setting

ro(c;) =log( I+ lc;l),k(c;) =(I+ lc;I 2 YI2 ,P(c;) =

c;k,k = 0, I,2, ... , and p = 2, we

arrive at the following result involving Sobolev space H'' obtained by Reider [81, p.884]

Corollary 6.4.3. Let f E H 5 (1R),s E IR, and lfl E H.B(JR)nL, (IR),/3 E No, with fJR.lfl(t)dt = I and Dklfl E L1 (IR) at least fork= {I, ... ,/3}. Then

lim~~~; LyY'f(·,a)- vCk ~df(-)11 s-k =0, a + 2

a--->0

where Ck = Cvkyt and Dk fork = 0, I, 2, ... , denote generalized derivatives.

6.5

Local convergence

In what follows we study the local convergence of the wavelet transform defined in local B~.k -space.

Definition 6.5.1 For n an open subset of IR, we define

called local generalized Sobolev space.

Lemma 6.5.2. f E (B~.k)toc(n) {::} f ·I/) E B~k 'v'I/J E Dw(n), where Dw(n) is the Bjorck space of C"' -functions of compact support inn defined by (6.2.6.)

Proof. Let f E

(B~k)Loc(n) and

1/J E Dw(n), then f agrees with an element of B~k on the

support of 1/J, hence 1/Jf E B~k· Conversely, suppose 1/Jf E B~,k for alii/) E Dw(n) and no is an open subset with compact closure inn. Choose 1/J E Dw(n) with 1/J =I on no, then f agrees with 1/Jf E B~k on no.


104

Definition 6.5.3. Let Un}nEN be a sequence in (B:,k)Loc(Q) and f E (B:,k)Loc(D.). Then {fn}nEN converges to fin (B~.k)Loc(Q) if and only if 111/>fn- fllp,k converges to zero for any If> E Dro(D.). Now, let us assume that

suppf = [-T, T] =I,

(6.5.1)

and consider (6.5.2) For 0

E D 00 (r>). For sufficiently large et/1 with 0 all

£

with et/1

~ £

< et/1 < T, we have supp(lf>)

~lefor

< T.

This implies that 1/>Td =If>! in B~.k(IR.) for et/1 ~

£

< T.

(iii) It is clear that both Td and fare in (B~,k)Loc(J2). Let a test function If> E De(J2) act on the distribution Tef.

(Te(t)f(t), 1/>(t)) = (f(t), Te(t)lf>(t)) = (f(t), 1/>(t)). This proves (iii).

Theorem 6.5.5. Let f satisfy (6.5.1) and (6.5.2). Let lfl be defined as in Theorem 6.4.2 and

Te as above. Then

) 12 LP(a-1D)lfi(Tef)(·,a)--+

~P(D)f

in

(B:,k)Loc(l~).

Proof. First we conclude that P(D)f = P(D)Td in (B~k)toc(J2). Let 1/> E Dro(J2). Then by the Leibnitz rule, we have

(P(D)Tef,l/>) =

L Ca

a;;.o

(t (~) i=O

1

(Da-if,DiTelf>))

105


a;;,o

=

([CaDaf,l/J)

= (P(D)f,cp). Therefore,

P(D)Tef = P(D)f. Now, let l/J E Sro(IR.) and T41: B~k ~ B~k be the multiplication operator defined by Trpf

=

(/J f. Then T41 is continuous. Let (/J E Dw(.fi)

c Sro(IR.). Then, using Theorem 6.4.2, we have 11(/J(·) al1/2LP(a-1D)Ifi(Tef)(·,a)- (/J(·) ~P(D)fllp.k/P

~IIT4111""11 a 1112 Lp(a-1DJifi(Td)(·,a)-

;,C P(D)(Tef)ll

v~d

_, pA/P

which completes the proof of Theorem. The next theorem provides the order of convergence and the method of proof is similar to that used in the proof of [81, Lemma 4.9]

Theorem 6.5.6. Let f be two times continuously differentiable in (b - e, b + e) for small

e > Oand bE R Let lfl E B~k(JR.)nL 1 (1R.) with fiR lfl(t)dt =land supplfl= [Tt, Tz]. Then for a > 0 sufficiently small, we have !r-f'(b) + O(a),

a- 312Lvlflf(b,a) =

vCt

where f' denotes ordinary derivative off, and Ct = CDifl· Proof. For small a, set [b+aTt,b+aTz]

=

I(b,a) and M(a) =

sup~E 1 If''(~)l. We have,

by mean value theorem,

la- 312Lvlflf(b,a)- !r,J'(b)l vC1

=

~

~ =

!r--11\lf/Df(b,a)- J'(b)l

vCt

!,! r

v C, a

llfl(t-b)llf'(t)-J'(b)ldt a

J1(b.a)

!r,!M(a) f

v C, a

K(a,b,J)a,

where K(a,b,J) = }JM(a) f~2 llfi(Y)IIYidy.

J1(b.a)

llfl(t-b)llt-bldt a

106

6.6


Example

As an illustration of the approximation property of the wavelet transform in Sobolev space H 5 we quote the following example from [81].

Let

l l

f () X=

1~

lxl

~ 1.5

2+x-I~x(y)dy i 1/n(x)(rxf)(y)dx

i =i =i =i =

=

1 1 g(y)dy 1 1 g(y)dy

1 1 g(y)dy

i i i

1 1 Pn(x)

(i

1 1 D(x,y,z)J(z)dz)

dx

1 1 J(z)dz.{ 1 Pn(x)D(x,y,z)dx

1 1 J(z)Pn(y)Pn(z)dz

1 J(z)Pn(z)dz 1

i

1 g(y)Pn(y)dy 1

(Pf)(n)(Pg)(n).

(7.3.32)

For various properties of Legendre convolution see [86]. (vi) Dual Poisson-Laguerre Convolution The dual Poisson-Laguerre transform [II] off E L(O,oo) is defined by

PL(f)(n) =F(n) =loco f(x)Lia)(x)d/\(x),

(7.3.33)

with the inversion formula

f(x)

=

(PL)-l [F(n)]

=

EF(n)da)(x)<J(n),

(7.3.34)

n=O

where

Lia) (x) = r(p )r( a+ I )da) (x), p(n) = n!fr(n +a+ I) e-xxa d/\(x)= r(a+I/x a(n) = [r(a + I)p(n)r 1• We set

D(x,y,z) = (PL)- 1 [da)(x)Lia)(y)](z) =

ELia)(x)Lia)(y)Lia)(z)<J(n);

(7.3.35)

n=O

(rxf)(y) =loco D(x,y,z)J(z)d A (z)

(7.3.36)

117


and

(!Ug)(x) =

~o=(rxf)(y)g(y)dA(y).

(7.3.37)

As in the previous case we get

PL(!Ug)

=

PL(f)PL(g).

(7.3.38)

(vii) Chebyshev Convolution Chebyshev transform off E L~ (-I, I) or C[ -I, I J is defined by ~[f](k) =

Ij' f(u)Tk(u)w(u)du,

~ = f(k)

1C

(7.3.39)

-1

where

Tk(x)=cos(kcos- 1 x), xE[-I,I], is the kth degree Chebyshev polynomial and w(x) =(I -x2) 112. The inverse Chebyshev transform is given by ~-

,~

I~

z=~

1C

1C k=l

[f](x) = - f(O) +-

L f(k)Tk(x).

(7.3.40)

Set (7.3.41) Then

j_', D(x,y,z)!(z)dz,

(7.3.42)

(!Ug)(x) = /_', ('rxf)(y)g(y)dy

(7.3.43)

~(JUg)= (~!)(~g).

(7.3.44)

(rxf)(y)

=

and

Conjecture 3.1. Let f E L~( -I, I) or C([-I, I]) and lxl ~ 1. Then for y E [-I, I],

(ry/)(y) =

~ {! ( xy+ V(I -x2)(I-y2)) + f

( xy-

V(I -x2)(I-y2))}

as given in [9, p.l66]. (viii) Sturm-Liouville Convolution Let I denote any open interval a

< x < b where a may be

-oo

and b may take the value oo.

Let { lfln (x)} ;=O be the orthonormal system of a Sturm-Liouville boundary value problem. Assume that the Sturm-Liouville transform off is defined by

S{f}(n) = F(n) =

1b f(x)lfln(x)dx.

(7.3.46)

118


Then (under certain conditions) inversion formula is 00

f(x) = s- 1 (F)(x) = L,F(n)1!fn(x).

(7.3.47)

0

For conditions of validity of the formula we may refer to [57], [98]. In this case we define

D(x,y,z) = s-l [l/ln(x)l/fn(Y)](z),

(7.3.48)

1 D(x,y,z)!(z)dz 1

(7.3.49)

(f~g)(x) = 1b('t"xf)(y)g(y)dy.

(7.3.50)

(-rxf)(y) = { and

7.4

Convolution for the wavelet transform

The above approach can be used in the development of a theory of convolution associated with the wavelet transform. For fixed and arbitrary p

~

0, and for any real number a =f. 0, the

continuous wavelet transform (CWT) of a signal f(t) E LZ(JR) with respect to the wavelet

q, E LZ(JR) is defined by (Wq,f)(b,a) =

l:f(t)~b;a(t)dt,

(7.4.1)

where

[D(x,y,z)](b,a)

=

l~ D(x,y,z)I/Jb;a(x)dx

= ll'b;a (Z )eb;a (y);

(7.4.5)

so that formally by (7.4.3),

D(x,y,z) = c;'l~l~ ll'b;a(z)Bb;a(y)I/Jb;a(x)lai 2P- 3 dbda. Clearly, D(x,y,z) is symmetric in all the three variables when

(7.4.6)

e = 11' = 1/J and 1/J is real

valued. The translation 't"x is defined by

('t"xf)(y) = f*(x,y) =

{~ D(x,y,z)J(z)dz

= Ci 1 l~l~l~ ll'b;a(z)Bb;a(Y)ll'b;a(x)J(z)lai 2P- 3 dbdadz.

(7.4.7)

(7.4.8)

Using (7.4.3) this can be expressed as (7.4.9) The associated convolution is defined by

(f~g)(x) = {~ f*(x,y)g(y)dy = {~~~ D(x,y,z)f(z)g(y)dzdy

(7.4.10)

= Ci 1 l~l~l~l~ ll'b;a(z)Bb;a(Y)I/Jb;a(x)J(z)g(y)iai 2P- 3 dbdadzdy. Using definition (7.4.1) we can write it as

(f~g)(x) = Ci 1 l~l~ (Wv,J)(b,a)(Weg)(b,a)I/Jb;a(x)lai 2P- 3 dbda = wl/>-l

[(W11,f)(b,a)(Weg)(b,a)](x);

(7.4.11) (7.4.12)

so that Wtf>[f~g](b,a)

= (W11,f)(b,a)(Weg)(b,a).

(7.4.13)

From representation (7.4.10), we conclude that if e = 11' = 1/J, then the following commutative and associative properties hold: (7.4.14)

120


(f~g)~h = f~(g~h).

Remark 7.4.1. D(x,y,z), 7:xf(y) and

(7.4.15)

f~g can be defined by other expressions similar to

(7.4.6), (7.4.8) and (7 .4.10) respectively. Then result of type (7 .4.13) will still hold. The following differentiability results can also be established.

For j,g E Y'(JR.), the

Schwartz space of rapidly decreasing functions, we have

where dxf(x)

D~(f~g)(x) = (Dk f~g)(x) + (f~Dkg)(x),k E No,

(7.4.16)

d~(f~g)(x)

(7.4.17)

= (dk f~g)(x) + (f~dkg)(x),k E No,

= x(d / dx)f(x).

Example 7.4.2. Let lfl(t)

· I 2 1 = B(t) = 0 and r > a+ I' then { lf/m,n} constitute a frame.

Now, let us define formally our basic generalized function D(x,y,z) (resp. D(x,y,z)) as follows:

D(x,y,z)

=

Wi 1 [lflm,n(x)lflm,n(Y)](z)

=

L lf/m,n(x)lflm,n(Y)lflm.n(Z), m,n=-oo

00

(7.5.5)

in case { lflm,n} form an orthonormal basis, and by

wi' [lflm.n(x)lflm,n(Y)](z) = L lflm,n(X)lflm,n(Y)ij/m,n(Z),

D(x,y,z) =

00

(7.5.6)

m,n=-oo

when {lflm.n} form a frame, provided that the series (7.5.5) and (7.5.6) converge. This will happen when Cm.n = lflm.n (x) lf/m,n (y) E 12 . Then

1:

D(x,y,z)lf/m,n(z)dz = lf/m,n(x)lflm.n(Y)

(7.5.7)

D(x,y,z)ij/m,n(z)dz = lflm.n(x)lflm.n(Y)·

(7.5.8)

and

1:

Next, we define the translation ('t' and i" resp.) by

(rxf)(y)

=

f*(x,y)

=

(i"xf)(y) = J*(x,y) =

j~ D(x,y,z)f(z)dz,

1:

D(x,y,z)f(z)dz.

(7.5.9)

(7.5.10)

Finally, we define the convolution

(f~g)(x) =

1:

f*(x,y)g(y)dy

(7.5.11)

122


and

(f"g)~(x) = j_~J*(x,y)g(y)dy

(7.5.12)

for the aforesaid two cases respectively. Proceeding as in the derivation of (7 .4.13) one can easily show that (7.5.13) and

Wiii(!Ug)(m,n) 7.6

= (Wijif)(m,n)(Wijig)(m,n).

(7.5.14)

Existence theorems

First we obtain boundedness results for the basic function D(x,y,z).

Theorem 7.6.1. Let (1

+ lwiP)cp(w) E U(IR), (J E U(!R), ~ + ~ =

1 and (1

+ lwiP)ljl(w) E

L 1(IR),p ~ 0. Then

ID(x,y,z)l ~ zP+~c;'IY- zl-b lx- zl-~-p liB llqll (1 + lwiP)cp(w) liP II (1 + lwiP)ljl(w) lit, where cl/> is given by (7.4.4).

Proof.We have

[D(x,y,z)[

~ c;'[f.f.ia[-P'I' (~) WPB (y:b )ta[-P~ ( x:b )ta[ p-3dadbl 2

[kk e~b) (y~b) (X~ ~ c;' kk IV'(u)IIB (y:z lief> (

= c;'

ljl

cp

(J

+u)

b

}a~-p-3dadbl

x:z +u) llai-P- 2 dadu.

Using Holder's inequality, we get

ID(x,y,z)l ~ c;'

k

lll'(u)ldu

(k le (y:z

+u) lq lal- 2da) ljq

X(k lcp (X: z + U) lp lal-pp- 2da )'jp

k

=

c;'ly-zl-' 1qlx-zl- 11p-pi1BIIq

x

(k lcf>(w)llw-uiPPdw) lfp

=

zPc; 11Y- zl-' 1qlx- zl-l/p-p II Bllq

lll'(u)ldu

klll'(u)

ldu

123


(

x

~ ltf>(w)IPiwiPPdw+ ~ ltf>(w)IPiuiPPdw ) I

I

1/p

I

= zP+-pC¢ 1 IY- zl_q_P lx- zl--p-P liB llqll (1 + lwiP)q>(w)IIP

x 11(1 + lwiP)q>(w)III· Theorem 7.6.2. (i) Let

VIE L 1 (JR),I/> ELP(!R), BE U(IR),p,q > 1,0 < p < 1 and~+~=

1 +p. Then

~ ID(x,y,z)ldz::;; C¢ 1C(p,p)lx- YI-p I V'IIIIII/>IIPIIBIIq,

(7.6.2)

where C(p, p) is constant.

VIE L 1(JR), (1 + lviP-l )tf>(v) EL 1(IR), (1 + lvlp-I )B(v) EL 1(JR), and p ~ I. Then ~ ID(x,y,z) ldz::;; c;Izp-llx- YI-p [111/>(v)vP-IIIIII Bill+ IIB(v)vP-IIIIIII/>11 I] I 'I'll I·

(ii) Let

(7.6.3) Proof.(i) We have

~ ID(x,y,z)ldz::;; C¢ 1 ~~~IV' ( z:b) lie (Y: b) Ill/> ( x:b) llai-P- 3dadbdz = C¢ 1 ~~ ie

(Y: b) Ill/> ( x: b) llai-P- a~ l'l'(t)ldtdadb 3

(7.6.4) Now, using Hardy-Littlewood-Sobolev inequality [47, p. 98] we get

IID(x,y,z)III where ~

+~

(ii) For p

~

:S:C¢ 1C(p,p)lx-yl-pii'I'IIIIII/>IIPIIBIIq, for O
(v)llviP-ldv ~ IB(u)ldu+ ~ IB(u)llulp-Idu ~ ltf>(v)ldv] C¢ 1 IIV'III2p-llx- YI-p

= c; lzp-llx- YI-p [111/>(v)vP-IIIIIIB III+ II B(u)uP-IIIIIII/>III]II 'I'll I· Theorem 7.6.3. (i) Let 1 +p. Then

VIE LP(JR), If> EL 1(JR), BE U(IR),p,q >

I ,0 < p < I and~+~=

124


EL 1(JR.), (1 + lulp-l )B(u) EL 1(JR.), (1 + lulp-l )lJI(u) EL1(JR.) and p ~ 1. Then ID(x,y,z)ldx ~ Ci 11Y- zi-p [illJI(u)uP-IIIIII eIII+ II B(u)uP-IIIIIIll'll d114'111· (7.6.6)

(ii) Let cp

k

The proof is similar to that of Theorem 7 .6.2. Theorem 7.6.4. (i) Let lJI EU(JR), cp

EU(JR), BE L 1(JR),p,q > 1,0 < p < 1 and~+~=

1 +p. Then

k

ID(x,y,z)ldy

~ Ci 1C(p,p)lx-zl-pllll'llqii4'11PIIBII1·

(7.6.7)

(ii) Let BE L1(JR.), (1

+ luiP-I )cp(u) EL1(JR.), (1 + lu1P- 1)lJI(u) EL 1(1R) and p ~ 1. Then ID(x,y,z)ldy ~ Ci 11x- zi-p [illJI(u)uP-IIIIIIC/' III+ llcf'(u)uP-IIIIIIll'll dII eIII· (7.6.8)

k

Next we obtain certain boundedness results for f~g.

EL 1(JR.), lJI ELP(JR), B EU(JR),p,q > 1,0 < p < l,f E L'(JR) and g E L1 (JR),r,r' > 1, ~ + fr +p = 2. Then llf~glll ~ c; 1C(p, p, r) IIC/'IIIIIll'llpll Bllqll!llrllgllr,

Theorem 7.6.5. Let cp

1, ~

+~

=

p+

(7.6.9)

where Ccp is given by (7.4.4) and C(p,p, r) is a constant. Proof. We have

kl(f~g)(x)ldx ~ k (k k k ~k k (k ID(x,y,z)ll!(z)ldz) lf*(x,y)llg(y)dy) dx

=

lg(y)ldy

lf*(x,y)ldx

lg(y)ldy

dx.

Therefore, by Theorem 7 .6.3 (i) we get

kl(f~g)(x)ldx ~

Ci 1C(p,p)II4'11111BIIqllli'IIP

=C- 1C(p,p)IIC/'IIIIIBII 1P

q

k

lg(y)ldy

llll'll f f P J'R.

}'R.

k

l!(z)lly-zi-Pdz

l!(z)llg(y)ldydz. IY- ziP

Now, invoking Hardy-Littlewood-Sobolev inequality [47, p.98], we obtain

In the sequel we shall use the following theorem whose proof can be found in [51, p. 75]. Theorem 7.6.6. Let f E L'(r > 1),0 1,0 < p
0.

Now, using Holder's inequality and applying (7 .6.11) again we get

llf~gllt ~ C¢ 1 2p+~ Kill/' llpllgllr' llfllrll (1 + lwiP)8(w) llqll (I+ lwiP)'I'( w) lit· Theorem 7.6.10. Let 1/J E L 1 (JR) nco(!R), 8 E U(!R), II' E U(!R), p, q

> 1, 0 < p < I, t +

~ = p + l. Assume further that f E U(JR),g E Lr' (IR), r, r' > I and~+~+ p = 2. Then Wq,(f~g)(b,a) =

Proof.

(Wvd)(b,a)(Weg)(b,a).

(7.6.18)

By Theorem 7.6.5., f~g E L 1 (JR). Therefore, for basic wavelet 1/J E L 1 (!R) n

co(JR), Wq,(f~g)(b,a) exists. Hence using (7.4.10) and (7.4.5) we can write

.k

Wq,(f~g)(b,a) = (f~g)(x)I/J ( x: b )iai-Pdx =

_k 1/J (x:b)iai-Pdx _k_kD(x,y,z)f(z)g(y)dzdy

=

.k

.k !(z)g(y )dzdy .k D(x,y, z)l/1 ( x: b) lal-p dx

= .k .k f(z)g(y)dzdyiifh;a(z) {Jb;a(Y) =

(Wv,f)(b,a)(Weg)(b,a).

7.7 An Approximation Theorem An approximation of g

E

U(!R) using wavelet convolution is given. For this purpose we

need the following theorem due to Okikiolu [52, p. 39].

128


Theorem 7.7.1. Let g E L'(JR),r >I. Then IIKa(g)- gllr----+ 0 as a----+ where Ka(g)

=~(~~and la,Q(a)

Theorem 7.7.2. Let L 1 (JR) ,f E

0+,

are given by (7.6.10).

(I+ luiP)IfJ(u)

L oo (JR) , and g E L' (lR), r

E U(JR),B E U(JR),~

> I.

+~

= 1,(1

+ luiP)lJI(u)

E

Then

lll(f~gp)l-lglllr----+0 as p----+1-,

where gp

=

ApQ(l-p)"

Proof. By inequality (7.6.15) and Theorem 7.7.1, we have

Remark 7.7.3. It is interesting to establish analogous existence and approximation theo-

rems for discrete wavelet transform.

Chapter 8

The Wavelet Convolution Product

8.1

Introduction

In this chapter we consider the n-dimensional wavelet transform defined by (3.1.1) and its variant (3.1.2). Using representation (3.1.2) a convolution associated with the wavelet transform is defined in terms of ordinary (Fourier) convolution ( 1.3.4.). In terms of the Fourier convolution, wavelet transform (3.1.1) can be expressed as follows: (8.1.1)

(W11,f)(b,a) = (/* Oa)(b), where

(8.1.2) Existence of convolution and some of its basic properties are summarized below; these will be used in the sequel.

II!* gil'

~

ll!ll,llgll'

(8.1.3)

[42, p.62]

and

(!*g)/\= jg [42,p.l58].

(8.1.4)

If f,g,h E L 1 (l~n), then we also have

(8.1.5) (ii) Iff E U(!Rn),g E U(!Rn),

i + ~ = l, l ~ p,q ~

oo,

then f

* g exists a.e.,

is bounded

and continuous, and

llf*glloo ~ IIJIIpllgllq 129

[42,p.64].

(8.1.6)

130


and (8.1.7) (iv) Let f E U(!Rn),g E U(!Rn), I ~ p,q

oo, 0 < J3 ~ 1.

(9.1.4)

s=O

We shall use the following definition of non-commutative convolution fog due to Riekstins [83] and the Laplace convolution f*g [46].

r/2

= Jo

f(x-t)g(t)dt

(9.1.5)

(f*g)(x) =lox f(x- t)g(t)dt.

(9.1.6)

(fog)(x)

and

The advantage of the convolution o over the convolution *·as noted by Li and Wong [46] is that in (9.1.5) f should be locally integrable on the open interval (0, oo) whereas in (9.1.6) it should be so on the semi-closed interval [0, oo ). Therefore fog may be defined for distribution fin Ltoc(O,oo) and distribution gin Lt;c(l~). the class of functions which are locally integrable on lR and which vanish on ( -oo,O). From (9.1.5) and (9.1.6) it follows that rx/2

(f*g)(x) =

lo

rx/2

f(x-t)g(t)dt+

Jo

g(x-t)f(t)dt

= (fog)(x) + (gof)(x).

(9.1.7)

We recall some of the properties off o g from [46, pp. 1541-1544]. Iff and g are (m- I)-times differentiable functions on (O,oo) and g E Lt;c(!R), then for x>O,

(! o!Y"g)(x)

=

1 m-1 Dm(f o g)(x) +D"'-l-f[f(x/2)d g(x/2)]. 2 f=O

L

(9.1.8)

The proof can be given by induction on m, cf. [46, pp.l541-1542]. In view of the property

o=

DH, from (9.1.8) we have

t-s-a oDj 0

for s, j E No and 0
0, b

lbl is Large

139

> 0 if x ~ 1, and b can be negative if x
I - a > 0. Furthermore, as in the above, using (9.1.14), we have

. -1-J I . looo (}+I) e-111 hm (t+ ,t/>1J(t)) = - 7 hm (Iogt)D 1 -(-)-dt 1)-->00 1! TJ--->0 0 t +z p (-I)i {"" Iogt = p(P)J+I Jo (t +z)P+}+I dt ( -1 )i r(p + j)

. z-P- 1 [Iogz-r-lfl(p+j)]· ' see the Appendix. Here ydenotes the Euler constant and lfl(z) = r'(z)/r(z).

= -j!

r(p)

(9.2.3)

We also note that

.

. looo e-111 f(t) looo ( f(t)) ( ) dt =

hm (!, t/>11 ) = hm

11-.o

t+z P o by Lebesgue's convergence theorem, where a+ p > 1. TJ->O

In view of (9 .1.24 ),

oo

o

Iim(fn,t/>1)) = Iim(-IY lo fn,n(t)JY; ( e

TJ-->0

TJ-->0

0

-TJI

) dt

t +z p

dt, t+z P

(9.2.4)

Asymptotic Expansions of the Wavelet Transform when

lbl is Large

143

=(-It{"" fnn(t)(-p)(-p-1)···(-p-n+l)(t+z)-p-ndt

lo

=

'

(P)n ~o= fn,n(t)(t+z)-p-ndt.

=lim 1)-->0

L,

j-1 (

-I)

1.

q=O

(9.2.5)

Tjj- 1-q(p)qz-p-q

q

= (p)j-1Z-p-j+ 1, j = 1,2, .... Substituting the values of (·, q,11 ) as 11

(9.2.6)

0 from (9.2.2), (9.2.6) and (9.2.5) into (9.1.28) we

---+

obtain

S ( )-n~1 f z -

. -p-j-a+1r(l-j-a)r(p+}+a-l) r( ) p

!-- a}z

J=O

n-1

(

)

+ L,(-l)j ~/M[f;j+l]z-p-j+Dn(z) j=O

where p > l -

(9.2.7)

].

a > 0, and Dn(z) = (P)n ~o= fn.n(t)(t +z)-p-ndt,

(9.2.8)

with fn,n(t) defined by (9.1.23). Also, if we substitute the values of(·,·) from (9.2.3), (9.2.6) and (9.2.5) into (9.1.30) and use the value of cj+ 1 from (9.1.31) we obtain the asymptotic expansion for a= I, _

Sj(Z) =

n-1

(-l)j

j=O

].

.

E aj-.,-(P)jz-P-J[logz- r- 'I'(P + j)] n-1

- L,cj+1(P)jZ-p-j+Dn(z), j=O

(9.2.9)

where On in the same as given by (9.2.8) and cj+ 1 by (9.1.31).

9.3

Asymptotic expansions for Tz and T3

The integral in Tz is the generalized Stieltjes transform of order p = s + f3 of f(v). Substituting the values of SJ(x) and Dn(x) from (9.2.7) and (9.2.8) in the series for T2 we derive T -~1n~1b -s-{3 -j-s-{3{ .r(j+s+a+f3-1)r(l-j-a) 1-a 2 - 1... 1... sY X a1 r( + f3) X

s=OJ=O

S

144


+ ( - 1)i (·~ + J3)j M[f;j + 1]} + Rn,l (x,y),

(9.3.1)

J.

where a + J3 > 1, 0 < J3 ~ 1, 0 < a < 1,

Rn.I(x,y) = Y!.bsY-s-~(x+J3)n {"" (

:n,n)~:~+ndu

Jo x u

s=O

(9.3.2)

and from (9.1.23),

fn,n(u)

=

( -1)n {"" (n-1)! Jo (-r-u)n-lfn(-r)d-r.

s

To find the asymptotic expansion of 1 (x) when (9.2.9) instead of (9.2.7) and get

Tz

(9.3.3)

a = I and 0 < J3 ~ I we apply formula

n-ln-1 [ (-1)1 =.?;Eo bsY-s-~ arJl-(s+ j3)j{logx+ 1J1(1 + j)- l!'(s+ J3 + j)} +cj+l (s + J3)J] x- j-s-~ + Rn,l (x,y),

(9.3.4)

where l!'(z) = r'(z)/r(z) and cj+ 1 is given by (9.1.31). Similarly substituting the asymptotic expansion of S8 (xy) we get T _n~ln~l 3 -

/.., /..,

s=OJ=O

a_,.y

-j-1 -j-s-a{br(j+s+a+f3-I)r(1-j-J3)( )1-~ X J r(s + a) xy

+( -1 )i (s +.~)} M[g; j +I]}+ Rn,z(x,y), 1· where a + J3 > 1, 0 < a ~ 1, 0 < J3 < I, n-1

f+a-1

Rn •z(x,y) = f;;Q ~ a.,.(s+ a)nY ( -1)n

gn,n(u) = (n- 1)! For 0
O,Re s

> 0,

(9.5.4)

where Dv(x) denotes parabolic cylinder function, we find that (r-a-1 q,) + '

X

IJ/2]+ I

(a) 1

k=O

(2

+ 1') 1. rka2k-2j-4 k! ( 2 + j - 2k) !

L (

2+}-2k -1 )i-m (2+J'-2k) b2+ j-2k-m am-a+ I m=O

X

E

3/2

= _ ~e-!(hfa) 2

m

f'(m- a+ 1)D_(m-a+l)( -bja).

Since, by [16, p. 267 (31)], Dv(x) = 2vf 2e-!x2lfl( -V /2, 1/2;x2 /2),

(9.5.5)

and by [ 16, pp. 262-263], lfl(a c· x)

''

=

1(1-c) + O(ixi 1-Rec) 0 < Re c < 1,x---> 0+, 1(1+a-c) '

(9.5.6)

we get t-a-j

(+

_

-i(b/a) 2 [i/2]+1 j-2k+2

,i/')-e X

k~ fo

f'(m- a+ 1) y'7irk-t(m-a+l)a2k-2j+m-a-3f2b2+j-2k-m

qm 2a + 1)

+ O(b3+}- 2k-m). Moreover,

( _ 1)i+m+l (j +2)! k!m!(2+}-2k-m)!

(9.5.7)

152


= a-2j-5f2e-lf2(bfa) 2

U/2]+1 (-I)k+j+l( .+ 2)'

L

1 1 • (ajV2,fkb-2k+j+2. • 1 k.(J +2- 2k).

k=O

(9.5.8)

In view of (9.1.24), error term: Rn

:= (fn,cfJ) = (-I)n {"" fnn(t)cfJ(n)(t)dt

lo

=

'

(-I)n+'fo''" !n,n(t)a3f2 (:rr+2 e-H'~b)2dt a-n-1/2 {co

=-

Jo

2n/2+1

'('-b)2

fn,n(t)e-2

a

Hn+2

(t-b) V'ia dt,

(9.5.9)

where fn,n is given by (9.1.23). Finally, from (9.1.28), (9.5.7) and (9.5.8) we obtain for fixed a E (O,oo), n-1 [j/2]+1j-2k+2

(Wv,f)(b,a) = e-!(bfa)2

L L L

j=O k=O

m=O

(-I )j+m+l (j + 2) !r(m- a+ I )y']r2-k-:l;(m-a+l) 1 k!(2+ j - 2k-m)!m!r(m 2a+ I)

x [a·~~~~~~~--~~~~~~----xa2k-2j+m-a-3f2b2+ j-2k-m + O(b3+ j-2k-m)] 1

-e-2(bja)

2

n-I[j/2]+1c· ,(-I)k+j+l(J'+ 2)'

L L

J+

j=O k=O X

.

k!(j+2-2k)!

2ka2k-2j-5/2b2+j-2k +Rn,

(9.5.10)

where Rn is given by (9.5.9).

Appendix

We demonstrate the evaluation of the integral

r

Jo

logt d p >0, (t +z)j+P 1'

used in (9.2.3). We know that

loo

co

1a ( ) +pdt = za+l-j-p B(a+ I,}+ p- a -I), t+z J

Differentiating formally with respect to

loo

co

talogt ( ) .+Pdt t +z J

a> -1, j

+ p- a- 1 > 0.

a we get .

= za+I-J-P [Iogz B(a+ 1,} + p- a +()daB( a+ I,}+p- a- I)]

-1)

Asymptotic Expansions of the Wavelet Transform when

lbl is Large

= za+l-j-p [logz B(a + 1,} + p- a- 1)

+ ( .1

rJ+P

) { r' (a+ I )r(j + p - a - 1)

-r(a+I)r'(J+p-a-1)}]. Now, let a

----+

0, then

{""'

logt. dt (t + z)i+P

lo

= zl-j-p [lo z r(j +p-I) g

r(j + p)

+r(J~p) {r'(I)r(J+p -I) -r'(J+p -I)} J 1-j-p

= .2

[logz-r-1Jf(J+p-I)], J+P- 1

where y is Euler's constant.

153

Chapter 10

Asymptotic Expansions of the Wavelet Transform for Large and Small Values of a

10.1

Introduction

Asymptotic expansion with explicit error term for the general integral

I(x) =

fo'"" g(t)h(xt)dt,

(l O.l.l)

where h(t) is an oscillatory function, was obtained by Wong [91], [93] under different conditions on g and h. Then the asymptotic expansion for (2.1.3) can be obtained by setting

g(t)

= eibt f(t)

for fixed bE JR. Let us recall basic results from [93] which will be used in

the present investigation. Here we assume that g(t) has an expansion of the form 00

g(t)rv [csts+A.-l as t---->0, s=O n-1 = Csts+A.-1 + 8n(t), s=O 1. Regarding the function h, we assume that as t

L

where 0 < A ~

( 10.1.2) ---->

h(t) = O(tP), p +A> 0, and that as t

---->

0+, (10.1.3)

+oo, 00

h(t) "'exp(inP) where -r

-I- 0 is real, p ~

I and 0
0+}0

(10.1.5)

This, together with (10.1.1) and [93, p.216], gives

n-1 I(x)= [csM[h;s+A]x-s-A.+Dn(x), s=O 155

( 10.1.6)

156


where Dn(x)

=

lim {"" gn(t)h(xt)exp( -EtP)dt. e-.0+ Jo

(10.1.7)

If we now define recursively h0 (t) = h(t) and h(-j)(t) = - ["" h(-j+ll(u)du, j = 1,2, ... , then conditions of validity of aforesaid results are given by the following [93, Theorem 6, p.217]:

Theorem 10.1.1. Assume that (i) g(m) (t) is continuous on (0, oo ), where m is a non-negative integer; (ii) g(t) has an expansion of the form (10.1.2), and the expansion ism times differentiable; (iii) h(t) satisfies (10.1.3) and (10.1.4) and (iv) and as t---> oo,t-f3gUl(t) = O(t-l-e) for j = 0, I, ... ,mandforsome E > 0. Under these conditions, the result (10.1.6) holds with (1 0.1.8) where n is the smallest positive integer such that It

+ n > m.

The aim of the present chapter is to derive asymptotic expansion of the wavelet transform given by (2.1.3) for large values of a, using formula (10.1.6). We also obtain asymptotic expansions for the special transforms corresponding to Morlet wavelet, Mexican hat wavelet and Haar wavelet.

10.2 Asymptotic expansion for large a In this section using aforesaid technique, we obtain asymptotic expansion of (W'I'f)(b,a) for large values of a, keeping b fixed. We have (W'I'f)(b,a)

=

'{! [~ eibwfii(aro)i(ro)dro

=

Va { {"" eibwfii(aro)j(ro)dro 2n lo

+

l""

e-ibw fii( -aro )j( -ro )dro}

= Va 2 7r (/1 +h), say.

(10.2.1)

Let us set

h(ro)

= l[i(ro).

(10.2.2)


157

Assume that 00

fi/( m) ""' exp( i-rmP) L brw-r-{3,

{3 > 0, m ---+ +oo,

't" =/:-

0, p ;:;::, I,

(10.2.3)

-r=O

and 00

f( m) ""'

L csms+A.- 1 as m---+ 0.

(I0.2.4)

s=O where 0 < A. ~ I. Also assume that as

m ---+ 0,

h(m) = fi/(m) = O(mP), p +A.> o. Then, as

( I0.2.5)

m ---+ 0, __

g(m) := eibw f(m)""'

"bw)r LCsWs+A.-1 L -~-,oo

oo

s=O

r=O

(

r.

_ ~ ~ (ibY s+A.-1+r - i...t i...t C5 - - W r! s=Or=O } m-'+A-1 (ib)r E Es ----;:y-Cs-r 00

=

{

s=O r=O 00

=

Ldsms+A.-1, s=O

(10.2.6)

where

d.,.

=

s (ibY -,-Cs-r· L r=O r.

(10.2.7)

For each n ;:;::, I , we write

n-1 g(m) = I'.dsms+A.- 1+gn(m). s=O

(10.2.8)

The generalized Mellin transform of h is defined by

r= r=

wz 1- 1h( m )e-ew dm M[h;zi] = lim e-->O+ lo wz 1- 1fi/(m)e-ewdm. = lim e-->O+ lo

(10.2.9)

Then by (IO.l.6),

n-1 h(a) = LdsM[h;s+A.]a-s-A. +8~(a), s=O

(10.2.10)

where (10.2.11) Also, from (I 0.2.5) we have

h(-m)=O(mP), W---+0, p+A.>O

(l0.2.I2)

158


and (10.2.13) Hence /z(a)

=

n-1

L,ds(-ly+A+IM[h(-ro);s+A]a-s-A. +D;(a), s=O

(10.2.14)

where (10.2.15) Finally, from (I 0.2.1 ), (I 0.2.1 0) and (I 0.2.14) we get the asymptotic expansion:

(Wvd)(b,a)=

't: {~ds(M[~;s+A.] + (-I y+A+I M [ fi/( -ro);s+

A]) a-s-A.+ Dn(a)},

(10.2.16)

where

r

r

Dn(a) = lim gn(ro)h(aro)e-E(J)dro+ lim gn(-ro)h(-aro)e-E(J)dro. E->O+lo E->O+lo

(10.2.17)

Since g(ro) = eibwj(ro), the continuity of j(ml(ro) implies continuity of g(ml(ro). Using Theorem

10.1.1

we get the following existence theorem for formula

(10.2.16).

Theorem 10.2.1. Assume that (i) j(ml(ro) is continuous on ( -oo,oo), where m is a nonnegative integer;(ii) j( ro) has asymptotic expansion of the form (I 0.2.4) and the expansion ism times differentiable,(iii) fi/(ro) satisfies (10.2.2) and (10.2.3) and (iv) as

ro

---7

oo, w-/3 j(j) (ro) = 0( ro-l-E) for j = 0, 1, 2, ... , m and for some e > 0. Under these

conditions, the result (10.2.16) holds with

(10.2.18) where n is the smallest positive integer such that A + n > m.

In the following sections we shall obtain asymptotic expansions for certain special cases of the general wavelet transform.

10.3

Asymptotic expansion of Morlet wavelet transform

In this section we choose

159


Fig. 10.1: Morlet wavelet (with

roo= 7).

25

2.0

1.5

1.0

0.5

-10

-5

Fig. 10.2: Fourier transform of Morlet wavelet (with

roo= 7).

Then from [14, p. 373] we have -(ro-"'o)2

iji( ro) = V2ice -----y- , which is exponentially decreasing. Therefore, Theorem 10.1.1 is not directly applicable, but a slight modification of the technique works well.

160


Assume that f has an asymptotic expansion of the form (10.2.4). In this case we have

h(ro) = ij/(ro) (10.3.1) and

h(ro)

= 0(1)

as ro----> 0.

(10.3.2)

Then from (1 0.2.1 0) and (1 0.3.1 ), we get

lt(a)

E

d.,.M [ J2ne -<w-zwo)2 ;s+A.] a-s-A.

=

+

!ooo gn(ro)J2ne -(aw-wo)2 e-ewdro,

Jim

(10.3.3)

£--->0+ 0

where

M [J2ne - 0,

where D-v(x) denotes parabolic cylinder function, we get

M [ J2ne - +=, {3 > 0.

( 10.6.3)

r=O Then

L brro-r-{3, 00

h( ro) ,...., eibro

b =/:- 0.

(10.6.4)

r=O Let 00

i/f(ro)rv[csros+A.-i, W---->0, OO.

(10.6.7)

Then proceeding as in Section I 0.2, we get n-1

It =c[csM[h;s+A.]c-s-A.+D~(c),

(10.6.8)

s=O where

r

(10.6.9)

= c L Cs( -l)s+A.+!M[h( -ro);s+A.]c-s-A. + s;(c),

(10.6.10)

D~(c) =

lim c if!n(ro)h(cro)e-Ewdro. E->O+ lo

Similarly, n-1

[z

s=O where (10.6.11)

Finally, from equations (I 0.6.8) and ( 10.6.1 0), we get

(W11,J)(b,a) =

':{! {1 cs(M[h;s+ A.]+ (-1)s+A+lM[h( -ro);s + A.])as+A.-i + Dn(a)},

(10.6.12)

where

Dn(a)

=

lim

r if!n(aro)h(ro)e-Earodro+ E->O+}o lim r if!n( -aro)h( -ro)e-Earodro. (10.6.13)

E->O+}o

!66


Theorem 10.6.1. Assume that (i) ijl(m)(m) is continuous on(-=,=), where m is a non-

negative integer; (ii) ijl( m) has asymptotic expansion of the form (1 0.6.5) and the expansion is m times differentiable; (iii) f(m) satisfies (10.6.3) and (10.6.6) and (iv) as m

----+ oo,

w-!3 ijl~j) ( m) = 0( w-l-e) for j = 0, I, ... ,m and for e > 0. Under these condi-

tions the result (I 0.6.12) holds with Dn(a)

=

(-l)mam j_~ ijl(m)(am)(eibwj(m))(-m)dm,

(10.6.14)

where n is the smallest positive integer such that A + n > m. Example 10.6.2. Let us consider asymptotic expansion for small values of a, of the Mexican hat wavelet transform discussed in Section 10.4. In this case

= V2ii(amfe-(aw)2/2

ijl(am)

= V2ii

f: (-1 yazr+z w2r+2 r=O n-1

2rr! a2r+2w2r+2 =V2iiJ;(-1Y 2rr! +o/n(am);

(10.6.15)

and j( m) satisfies (10.6.3) and (10.6.6). Then, following the technique of proof of ( 10.6.12) and using formula [DkfY'

(W111.f)(b,a)

= (im/f(m),k E No [57, p.

39], it can be shown that

J;

1 n-1 ( 1)r { ;r! M[eibwj(m);2r+3]

= ..j2n

+ M[e-ibwj( -ro);2r+3] }a2r+S/Z + Dn(a) =-

V2iifa 2;r!D2r+2 .f(b)a2r+5/2 + Dn(a),

(10.6.16)

where Df denotes generalized derivative off and

Dn(a)

=,;a lim r eibw i( (1)) 1/fn (am )e-E(J) dm 21C E-->0+ JE-->0+ (10.6.17)

Example 10.6.3. Asymptotic expansion for small a of the Morlet wavelet transform can be obtained as follows. Here

lj/(m) = V2iie-(w-~)2/2

= V2iie-wJ;z

E(-1Ym2s E(~my s=O

2ss!

r=O

r!


= ..fiie-w1il2

167

E

dso.i

s=O

~ dsro s + =-----( vr;:c L.ne -w1J/2 1... l!'n ro ) ,

=

(10.6.18)

s=O

where [s/2]

ds =

(_I )i (J.)s-2}

L 21 J.. s }=0 1(

_ 02

Assume, as in the previous example, that

.) 1 by [80,p.57(7)].

(10.6.19)

1 .

j( ro)

satisfies (I 0.6.3) and ( 10.6.6). Then, pro-

ceeding as in the above example, we find that (10.6.20)

n-1 ..fiie-ro&/2

=

L dsas+lf2e-isnf2(Ds f)(b) + Dn(a),

(10.6.21)

s=O

where (10.6.22) Example 10.6.4. Consider now the Haar wavelet transform. In this case

(10.6.23) Then we find that

(Wy,f)(b,a)

=

E

y'a {n-1 ·r+l ( I ) 2; l r! I - 2r-l M[eibw j(ro);r]

+( -I)r-1 M[e-ibw J( -ro); r]} ar-1 + Dn(a) = - nE r=2

~(I- 2r~l) vr-lf(b)ar-1/2 + Dn(a),

(10.6.24) (10.6.25)

where (10.6.26)


168

10.7 Asymptotic expansion for small a continued In this section we obtain asymptotic expansion of the wavelet transform given in the form (9 .1.1) when a

-+

instead of j and

0+. Naturally, in this case we have to impose conditions on

f

and 11'

fi/.

Now, Jet us write (9.l.l) in the form:

(Wy,f)(b,a) = c 1 / 2 [~ f(t + b)l!'(ct)dt, where c

(10.7.1)

= I I a -+ +oo and b is assumed to be a fixed real number.

Then setting g(t)

= f(t +b)

and h(t)

(Wy,f)(b,a)

= c 112

= l!'(t), we have

[fooo g(t)h(ct)dt+ [ 0 g(t)h(ct)dt]

= c 112 [lt

oo

+h] (say).

(10.7.2)

Assume that g(t) satisfies (10.1.2) and h(t) satisfies (10.1.3) and (10.1.4). Then from (10.1.6) it follows that

n-l It = I:CsM[lji;s + l.]c-s-A. s=O

+ 0~ (a),

(10.7.3)

where

O~(a) =

r= 8n(t)o/(tla)e-EtP dt;

Jim

e-.o+lo

(10.7.4)

and n-1

h =

L Cs( -l)s+A.-l M[ 11'( -t);s +A.]+ o;(a),

(10.7.5)

s=O where

o;(a)

=

Jim

r= 8n( -t)l!'( -tla)e-EtP dt.

e-.o+lo

(10.7.6)

From (10.7.2), (10.7.3) and (10.7.5) we get

n-l (W'I'.f)(b,a)

=

L c M[lji;s+l.]a +A.-If2 5

5

.1'=0

n-l

+ L c5 ( -l)s+A.-l M[l!'( -t);s + A.]as+A.-l/2 + On(a),

(10.7.7)

s=O where

On(a) = a- 1/ 2 { lim

r= 8n(t) IJI(t I a)e-et'' dt + e-->0+ lim r= 8n( -t) 11'( -t I a)e-etP dt}. lo

e-.O+ lo

(10.7.8)

169


Example 10.7.1. Let us find again asymptotic expansion of Morlet wavelet transform for small a, using the above technique. Here

lf!(t)

=

eiWot-t 2/2.

Suppose that g(t) = .f(t +b) satisfies (1 0.1.2). Then from (10.7.3), using (9.5.4), we get n-1

h = [. CsM[lJi;s+ A.]c-s-A. + D~ (a) s=O n-1 =

L Cs Jo{"" eiWot-tz/2ts+A.-1dtas+A. + D~ (a)

s=O

0

n-1

= [. c5 r(s+A.)e-cofii4 D_ 5 _A.( -i~)as+A. + D~ (a).

(10.7.9)

s=O

Similarly, n-1 = [ . cs( -Iy+A.- 1r(s

h

+ A.)e-cofii4 D_s-A. (i~)as+A. +

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