Applied mathematics reviews. Volume 1

Applied Mathematics Reviews Volume 1

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Applied Mathematics Reviews Volume 1

Editor

George A Anastassiou The University of Memphis, USA

V f e World Scientific «■

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pie. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

APPLIED MATHEMATICS REVIEWS VOL.! Copyright © 2000 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-4339-1

This book is printed on acid and chlorine free paper. Printed in Singapore by World Scientific Printers

Dedicated to my daughters, Angela and Peggy

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PREFACE

This volume contains fourteen high-quality reviews of main subareas of Applied Mathematics. These are survey articles containing also original results, written by top people of the field from around the world. Applied Mathematics plays a central, very active role in contemporary science, shaping our lifes better. Feeling that this volume is a very important research tool to the use of Applied Mathematicians, we intend to publish in the future subsequent volumes. In the following we would like to mention the important topics that are reviewed in this volume: convergence of generalized singular integrals, com­ putational concepts in numerical analysis, approximation in modular spaces, simultaneous approximations, minimal projections, copositive approximation, neuro-fuzzy adaptive control, sampling theory, stable financial modeling, per­ formance measurements in finance, evaluation of statistical functionals, dy­ namical systems for the solution of nonlinear ill-posed problems, extrapolation in partial differential equations, and interpolation and approximation with wavelets. This applied mathematics review volume is suitable for libraries in the mathematical sciences, engineering and economics, also good for individual re­ searchers and graduate student,s (classroom and research) of the related fields. The editor would like to thank Mrs. Kate MacDougall of Warren, R.I, for the excellent, prompt typing job preparing this volume. The Editor George A. Anastassiou University of Memphis, Memphis, TN February 1, 2000

VII

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CONTENTS

Preface

vii

Convergence of Generalized Singular Integrals to the Unit, Multivariate Case George A. Anastassiou and Sorin G. Gal

1

Two Contemporary Computational Concepts in Numerical Analysis Ioannis K. Argyros

9

Linear Integral Operators with Homogeneous Kernel: Approximation Properties in Modular Spaces. Applications to Mellin-type Convolution Operators and to Some Classes of Fractional Operators Carlo Bardaro and Ilaria Mantellini

45

On the Simultaneous Approximation of Functions and Their Derivatives Theodore Kilgore

69

On the Computation of Minimal Projections: Millennium Report B. L. Chalmers, F. T. Metcalf, and B. Shekhtman

119

Copositive Polynomial Approximation Revisited Ymg Kang Hu and Xiang Ming Yu

157

Neuro-Fuzzy Adaptive Control: Structure, Algorithms, and Performance Augustine 0. Esogbue

175

Sampling Theory and Function Spaces Hans-Jiirgen Schmeisser and Winfried Sickel

205

Computational Issues in Stable Financial Modeling Carlo Marinelli and Svetlozar T. Rachev

285

Performance Measurements: The Stable Paretian Approach G. Gotzenberger, S. T. Rachev, and E. Schwarz

329

Evaluating Statistical Functionals by Means of Projections onto Convex Cones in Hilbert Spaces: Part I & II Thomas Rychlik

407

IX

Dynamical Systems and Discrete Methods for Solving Nonlinear Ill-Posed Problems Ruben G. Airapetyan and Alexander G. Ramm Extrapolation: From Calculation of 7r to Finite Element Method of Partial Differential Equations Xiaoping Shen A Survey on Scaling Function Interpolation and Approximation En-Bing Lin List of Contributors

x

491

537 559 609

C O N V E R G E N C E OF GENERALIZED SINGULAR I N T E G R A L S TO T H E U N I T , MULTIVARIATE CASE George A. Anastassiou and Sorin G. Gal Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA E-mail: [email protected], [email protected] Department -of Mathematics, University of Oradea, Str. Armatei Romane 5, 3700 Oradea, Romania E-mail: [email protected] In this article we study the degree of Lp-approximation (1 < p < + oo) to the unit, by multivariate variants of the Jackson-type generalizations of Picard, GaussWeierstrass and Poisson-Cauchy singular integrals. These results are extensions of those in the univariate case proved by the authors in [2].

1

Introduction

Let / be a function defined on R m with values in R. Let x = (xi,... ,xm), h = (hi,... ,hm) € R m . Let us denote

^

t=0

'

We define the rth L p -modulus of smoothness over R m , 1 < p < -f-oo, by Wr (/; 0, i = hm. Reasoning exactly as in the univariate case in [2, proof of Theorem 2.1], (5) implies (for 0 < & < 1, i = - l,m) ll/-^n.€(/)IU


11/ - Q„.«(/)IU
Wn,f(/) -~> / ) , for the generalized operator Qn^(f) in general this does not happen, because when & —» 0, we have Kp(n, &) —► +oo, for all 1 < p < +oo. However, under some smoothness conditions on / , (as for example if 31*1/ 1 < M on R m , for all \k\ € {0, l , . . . , n + 1}, where \k\ l dx\ ...dx$ ki + ... + km, ki € N U {0}, i = l,m) and reasoning as in the univariate case [2], we easily obtain that Pn,t{f) —» / , as £ -+ 0. References 1. Anastassiou, G.A. and Gal, S.G., Global smoothness preservation by multivariate singular integrals, Bull. Austral. Math. Soe.(to appear). 2. Anastassiou, G.A. and Gal, G.S., Convergence of generalized singular integrals to the unit, univariate case, submitted.

8

T W O C O N T E M P O R A R Y COMPUTATIONAL A S P E C T S OF N U M E R I C A L ANALYSIS

Department

loannis K. Argyros of Mathematics, Cameron University, Lawton, E-mail: [email protected]

OK

73505

In this study we report advances in two computational aspects of Numerical Anal­ ysis. In Part 1, we use inexact Newton-like iterates to approximate a solution of a nonlinear equation in a Banach space. Solving a nonlinear equation using Newton­ like iterates at each stage is very expensive in general. That is why we consider inexact Newton-like methods, where Newton-like equations are solved only ap­ proximately and in some unspecified manner. In the second part we examine the connection between inexact Newton-like methods and their discretized versions. In particular we show that the behavior of the discretized process is asymptotically the same as that for the original iteration and consequently, the number of steps required by the two processes to converge to within a given tolerance is essen­ tially the same. Our results compare favorably with earlier ones. Some numerical examples are also provided to show that our results apply where others fail.

1

Part 1

1.1

Introduction

We are concerned with approximating a solution x* of the equation F(z) = 0,

(1)

where F is a nonlinear operator defined on a Banach space E\ with values in a Banach space E2 with the properties: (a) there exists a point x* e Fi with F(x*) = 0; (b) F is m-times continuously Frechet-differentiable in a neighborhood of x* where m is an integer with m > 2; (c) F'(x*) is nonsingular; Note that F'(x') ~E L(E\, F 2 ) the space of bounded linear operators from E\ into £2 denotes the first Frechet-derivative of F evaluated at x = x*, whereas F ( m ) ( x ) 6 £(£{", F 2 ) (9], [17] is the m-th Frechet-derivative of oper­ ator F. Inexact Newton method is any procedure which, given on initial guess yo, generates a sequence {yn} (n > 0) of approximation to y* as follows: For n = 0 Step 1 until convergence DO. 9

Find some step xn which satisfies F'{yn)xn

= -F(yn)

+ r„,

(n > 0)

(2)

where

8 Set yn+\ =--yn+xn

(n>o).

(4)

The numbers r?„ depend on y„ (n > 0). In particular for r/n = 0 (n > 0) we obtain Newton's method [2]-[9], [16]-[20]. In the elegant paper [13] the local behavior of inexact methods is analyzed in the special case when E\ = E2 = R ' (i € N) However, instead of condition (b) above the same assumption but on the first Frechet-derivative is made. We observed that under our assumption all results obtained in [13] also hold under our hypotheses. Moreover, we observed that our upper error bounds on the distances involved are smaller than the corresponding ones in [10], [13]. This is due to the passing from the first to the mth Frechet-derivative. Such an approach has produced sufficient conditions for the convergence of Newton's method to a locally unique solution of Eq. (1) different from the classical Newton-Kantorovich hypothesis [4], [9], [16], [17]. In particular we show: inexact Newton methods are locally convergent if the forcing sequence is uniformly less than one; we show how to choose a forcing sequence which preserves the rapid convergence; of Newton's method. Finally, in Remarks 1 and 3 our results compare favorably with the corresponding ones in [11], [13]. 1.2

Convergence Analysis

We will need a result on the mean value of mth Frechet-differentiable operators. R e m a r k 1. It follows from the hypotheses that there exists constants d, Q{ ( 2 < i < m , m > 2 a n integer) such that ||FV)-1li

and

a, > I I F ^ V ) ! !

Consider the approximations for p := y-y*,

(2 < t < m).

6i := y* + #iP, h :- y * + ^ ( b i _ 1 -

y*),fte[o,i]: F'(y) - F'(y*) = F'(y) - F'(y*) - F"(y*)(y - y") + F"(y*)(y - y') = f {F"(y'+6lp)-F"(y')}d81p+ Jo

f F"(y*)pde1 Jo 10

(5)

.1

=

/■!

/ / F"(b2)(b1-y*)pd02dei+ Jo Jo

1 F"(!/*)pd0, Jo

= ■■■=[■■■{ [F^(bm-i)-F^(y')](bm-2-yl Jo Jo • ■■(bi-y')pdBm.i---M2dOi F<mHy')(bm-2-y')-(bl-y*)pd6rn-1dem-2--d92d6l

+ [ ■ [ Jo Jo +

F<m~1'>(y')(brn-3-y')---(bl-y')Pdern-2---d62de1

f ... f Jo Jo

+ ■■■+ [ F'\y')pd6,-

(6)

and for a\ — y* + 6\p, a, = y* + #i(a,_i - ym), i = 2,... ,m - 1 as in (6) we have: F(y)-F(y')-F'(y')(y-y-)

= [ ... f [F^(am^) - F^(y')}(am.2 - y') Jo Jo ■■■{al-y'){l-6i)pd6m-1---dB2dOl

F(mHy*)(am_2-y*)--(al-y*)(l-e1)pdem-l'--d62d81

+ [■■[ Jo Jo

+ ■■■+[ F " ( y ' ) ( l - ^ ) P 2 ^ 1 . (7) Jo Since F' is continuous in a neighborhood of y*, for given do < d, by (6) there exists 60 > 0 so that \\F'(y) - F'(y')\\ < do, whenever y e U(y*,6o) — {y € E\ | \\y - y*\\ < bo)- It follows from the Banach Lemma on invertible operators [17] that F'(y) is invertible and WF'iyr'W^du

(8)

where, d\ = d(l — ddo)~l- Moreover, we obtain \\F'(y')'1

~ F'(y)-l\\

= \\F'{y')-\F'{y) F'(y*)}F'{y)-l\\ 0 such that \\F)

< \\yk -y*\U + [ ■2T + ¥ I I y * " y \\yk - y

m!

•II2

(25)

By (23), (24), and (25) estimate (22) gives l|yk+i - y ' l l . < (i

+M)\vn

+ --- +

+ gj-llj/fc-y

q

m!

9

+ (m -

Y

\\yk -y II.

1)!

-^m-\\yk-y-\r-2)\\yk-y'"2

Hl/fc - y*\\n + TZT^WVk (m-l)!'

+ l f + ^\\yk-y'\\ +

q + otn

+

ml

*i|2

- y*\\m + ■ ■ ■ + a2\\yk - y 2

■ l | y * - y l m - 2 ))llj/ l l y f cf c--y*ll y*"2

< (1 + nq)t\\yk - 3/-1U + (1 + M)t\\yk - y'f ( ^ + a2 + g ) + ( 1 + M ) % f c -y*f^+a3 + ... + (l +

+ ^) 1

(1q)t\\yk-y'\r-

+ (i + M ) % k - y * l

m!

am-\

,

,

Ckm-1

( m - l ) ! + am_i + ( m - l ) ! 0, respectively. Note that we have also assumed (without loss of generality that (18)-(20) above and (2.3)-(2.5) in [13] hold for the same e and q. It then follows from (28) and (29) that our error bounds (14) are better than (2.1) in [13] provided that (30)

tA < tT.

By (13), (26) and (27) it follows that (30) is true if ||y0 - 2/1 sufficiently small Since then (30) is true if (31) to < tr, which is true. The following lemma is crucial. L e m m a 1. For any e > 0 there, exists 6 > 0 such that: / / . . * \ —l II —l

\F'(y')

+ ••• + m!

< {ll*"(y' + ••• + —f < ||F(j,)|| < + ••• + - ; if,

Q 2

,

2T 12/ - V

¥

y-v

Q 3

+

II

~Uh~y

*|im-2

+

l } l y-y

¥ll?/-2/l

lim-2

\\F'(y*)\\ + y-y

112/ - y

»|im-2

-2/l}ll2/ Q 2

2f

+

i

Q3

II

3flly_y (32)

y'\\}\\y-y'\

\\y-y*\\ 0 there exists S > 0 sufficiently small so that (10) is satisfied if ||y -y*\\ < 6. Since, F(y) = F V X i / - !/*) f {F(y) - F(y') - F'(y')(y 15

- y')},

(33)

we get by (10) and (33), \\F(y)\\{\\F'(y-)\\-[^ +- +

+

^\\y-y*\\ v-y*\\}\\v-ym\

*|im-2

-l\w-y

{ l l ^ r M l ^ - I f + llly-y'll

>

.» IITTT — 2

+ ...+ — ||j/-y

112/ — 2/

}lli/-y*l

li\\y-y'\\ 0 such that: (34)

\\F^Hy)~F^(y')\\ 0) be a sequence which converges to y*. Then, (a) y„ -» y* superlinearly if \\yn+1 - y"\\ = 0{\\yn - y'\\) as n -> oo; (b) yn —> y* with strong order at least r (r > 1) if ||y„+1-yl=0(||y„-yT)

as n - co;

(c) y„ —> y* with weak order at least r (r > 1) if limsup \\yn - y*\\l^r

oo.

n—*oo

Note also that given continuous nonnegative real-valued functions f\ and f2, we write f\{Vn) = 0(f2(yn))

a s n - t o o , if limsup ' n-»oc

" = 0,

J2\yn)

and fi(Vn) = S(f2(yn))

as n -> co, if limsup n-*oo

l

" < +00.

J2\.yn)

Theorem 2. Assume inexact Newton iterates {y„} (n > 0) converge to y*. 77ien, y n —> y* superlinearly if and only if \\rn\\ = 0(\\F(yn)\\) 17

asn^w.

Morever, if F ' m ' is Holder continuous with exponent X € (0,1] at y*, then !/„->!/' as n —> oo with order at least 1 + X if and only if Iknll = &(\\F(ynW1+X)

asn-oo.

Proof. The proof is similar to Theorem 3.3 [13, p. 404], but there exist nontrivial modifications. At first, assume yn —» y* as n —> oo superlinearly. By the approximation rn = [F(yn) - F(y') - F'(y*)(yn - y*)\ + [F'(yn) - F'(y*)}(yn - y*) + [F'(y') + F'iVn) - F'(i/*))](y n+1 - y').

(36)

taking norms and using (23) and (24), we obtain l|r„|| < ||F(y n ) - F ( y ' ) - F'(y*)(yn - y*)|| + ||F'(y n ) - F'(y*)ll \\yn - y'\\ + [ l | i r V ) l l + l|i 7 , '(2/r 1 )-F'(y-)||]||y n + 1 -y*|| < 0 ( l l y n - y * l l ) + 0(i)Hyn-y*ll + [||FV)ll+0(l)]0(||yn-yl).

(37)

Hence, by Lemma 2 we get llrnll = 0(||y» - y*||) = 0(\\F(yn)\\) as n - oo. Conversely, assume ||r n || = 0(||F(x n )||). By (21) we also get

||yn+1 - y'|| < [HFV)-1!! + ||F'(yfc)-1 - ^V) - 1 !!] •[lkn|| + ||F'(y n )-FV)llll?/n-y*|| + ||F(y n )-F(y*)-FV)(y»-y')ll] = [|I^V)- 1 H+0(l)-[0(||F(y n )||) + 0(l)\\yn-y'\\

+

0(\\yn-y*\\)\.

Hence, by Lemma 2 we get |jy n + 1 - y*|| = 0(\\F(yn)\\)

-f 0(|jy n - y'||) = 0(\\yn - y*||)

as n ^ oo.

(m)

If F is Holder continuous with exponent A at x*, then using (35) instead of (10) the proof is essentially the same. That completes the proof of Theorem 2. □ The following result give a characterization of weak order of convergence. Theorem 3. Assume that inexact Newton iterates {yn} (n > 0) converge to y*. If F ' m ) is Holder continuous with exponent X € (0,1] at y*, then yn —► y* 18

as n —► oo with weak order at least 1 + A if and only ifrn—*0 weak order at least 1 + A.

as n — ► 00 with

Proof. The crucial estimates (3,1)—(3.5) in Theorem 3.4 [13, p. 405] hold here also by (32), (8), (9), (34), and (35). Hence, the proof of our theorem folows. That completes the proof of Theorem 3. □ The following result relates forcing sequences and inexact Newton iterates. Corollary 1. Assume inexact Newton iterates {yn} {n > 0) converge to y*. Then (a) yn -» y* superlinearly if lim„_ 0 0 nn = 0; (b) yn —* y* with order at least 1 + A if F" is Holder continuous with exponent A e (0,1] at y* and Vn = 0(\\F(yn)\\x)

asn^co;

(c) yn —♦ y' with weak order at least 1 + A i / F ' m ' is Holder continuous with exponent A £ (0,1] at y* and {rjn} (n > 0) converges to 0 with weak order at least 1 + A. Favorable comments for our Theorems 2 and 3 when compared to the corresponding Theorems 3.3 and 3.4 in [13] similar to the ones made in Remark 1 after Theorem 1 can now follow. Another convincing result is given by the following: Theorem 4. Let F : D C E\ —» E2 be an operator such that: (a) there exists ay* e D with F(y*) = 0; (b) F is m-Frechet-differentiable in a neighborhood of y* contained in D; (c) F'{y*) is nonsingular on D; (d) there exists nonnegative constants a, a*, 2 < i < m, c, d, h, 7 such that: :\F(m)(y) - Fim)(y*)\\

< a\-,y - y"jl+1

for \\y - y*u sufficiently small. (38)

ll^(l)(y*)ll < «n IIF'Q/T1!!^, HFV)II < h,

(39) (40) (41)

and llrnll^cllF^)!!2 19

(n>0).

(42)

Then, there exists e > 0, such that ifyo S U(y',e) = {y 6 E\ | ||y — y*|| < e), then inexact sequence {yn} (n > 0) generated by (4): yn —> y* quadratically in the sense that lbn+i-?/l0),

where (2a£^+22l 2 am _ m - 2 Co — d + — ^ £ " ' " ' + •••+02 \ (3 + 7 )m! m! m+7

+ 2c

m-l

(2 + 7 ) ( 3 + 7 )m!

m!

2!

Moreover, the following estimate holds: (44)

■^T^F**-*-**)-

Proof Choose £ > 0 so that F'(y) is nonsingular for all y € U(y*,e). Let e > 0 be such that m+

~>-1 2amem~2 \(3 +7 ) m ! + — ^ T ~

J(2ae d

+

--

+ Q 2

a£m+7 a m £- 'm" -- l' a2 2c|/l+ + + + ( 2 + 7 )(3+ 7 )m! ^^r- '-- ^ ! £ „ .,

+

K

(45)

and .F ( m ) is Lipschitz continuous satisfying (38) in U(y*,e). Using ||yo -y*\\ < e, (45), (38)-(41) and the approximation (6) multiplied by F'(y*) _ 1 we get \F'(y')-llF'(yo)-F'(y*)}\\ oo, estimate (44) follows from (43) by letting e —0. That completes the proof of Theorem 4. □ Remark 3. Let F' be Lipschitz continuous in a neighborhood of y* contained in D with constant ao- Then Theorem 2.2 in [10, p. 409] or Theorem 3.3 in (13, p. 404] give error estimates ||2/n+i - 2/1 < d a0 + 2c (h +

^ )

Il2/n-y*|| 2

(n>0),

(50)

< fa = d(a0 + 2ch2) = 0O + d{a0 - Q 2 ) .

(51)

and limsup ll.f"+1

rwoo

*J

\\yn-y\r

However, if for example Q 2 < ao, then our estimate (44) is an improvement over (51) (see also the example that follows). Remark 4. As in [10], [11], [13] the results obtained here can be used for pro­ jection methods such as Arnoldi's method, the generalized minimum residual method (GMRES), the generalized conjugate residual method (GCR), and for combined inexact-Newton/finite-difference projection methods. Remark 5. The results obtained here can also be used to solve autonomous differential equations [9], [17] of the form F'(x) = P(F(x)),

(52)

where P : Ei —> E\ is a known continuously Frechet-differentiable operator at y*. Since F'(y*) = P(F(y*)) = P(0),F"(»*) - F'{y*)P'{F{y*)) - P(0)P'(0), we can deal with the results obtained here without actually knowing the solu­ tion y*. Example 1. Let E\ = Ei = R, and consider the functions F, P defined by F(x) = ez - a,

P(x) = x + a,

a > 0.

Then, functions F, P satisfy (52), F'{y*) = a = h = d~\ F"{y') = a = a2, y* = lna, and for y € U(y*,e), e > 0 we get ao = ae £ > a = a 2 . Hence, our upper error bound (44) is smaller than upper bound (51). 22

2 2.1

Part 2 Introduction

The goal of this part is to extend the validity of the mesh independence prin­ ciple for inexact Newton-like methods initiated in a previous paper [8]. Let us consider the problem of approximating a locally unique solution x* of the equation F{x) + Q(x) = 0, (53) where F, Q are nonlinear operators defined on some closed convex subset D of a Banach space E\ with values in a Banach space Ei. Let x 0 € D be fixed and define the inexact Newton-like method for all n > 0 by yn = xn-A(xn)-l(F(xn) + Q(xn)) (54) and £n+i =yn-

zn.

(55)

Here, A(xn) denotes a linear operator which is an approximation to the Frechet derivative F'(xn) of F evaluated at x — xn (n > 0). The points zn € D for all n > 0, and are determined in such a way that the iteration {xn} (n > 0) converges to a solution x* of Eq. (53). The convergence of both methods have been examined extensively by others and us, [1], [2], [5], [8], [14], in connection with the mesh independence principle. Since the iterates of the inexact Newton-like method (54) -(55) (for zn = 0, n > 0 or not) can rarely be computed in infinite dimensional spaces, (54)-(55) can be replaced in practice by a family of discretized equations P{a) = 0

(56)

indexed by some real number h > 0, where P is a nonlinear operator between finite dimensional spaces E1 and E2. Let the discretization on E\ be defined by the bounded linear operators L : E\ —> E1. Consider also as in [8] the iteration {an} (n > 0) given for all n > 0 by bn = an - S(an)~lP(an),

aQ - L(x0)

(57)

and a-n+i =bn-dn.

(58)

here S(an) denotes a linear operator which is an approximation to the Frechet derivative P'(an) of P evaluated at a = an for all n > 0. The points dn e El for all n > 0, and are determined in such a way that the iteration {an} (n > 0) 23

converges to a solution a* of Eq. (56). Note that all symbols introduced in (56)-(58) really depend on h. That is P = P/,, L = L/,, 5 = S/i, etc. But we do not use the latter to simplify the notation. The mesh independence principle (proved in the studies listed above) as­ serts that the number of steps required by the two processes to converge to within a given tolerance is essentially the same. Here we show that this is true for our inexact Newton-like method (54)-(55). In contrast with [8] we consider a nonuniform discretization and use norms corresponding to the domain of defi­ nitions of the operators. Our results can be reduced to the ones obtained earlier for appropriate choices of the factors involved. We make use of the sufficient conditions for the convergence of the inexact Newton-like methods (54)-(55) that we found in [8]. The importance of the formulation of an efficient mesh size strategy based upon the mesh-independence has been extensively discussed in [1], [2], [5], [8], [14] and the references there. We finally apply our results to solve a two point boundary value problem. 2.2

Preliminaries

The norms in all spaces will be denoted by the same symbol || ||. For any bounded linear operator from E\ to Ei or from El to E2, the induced norm will be used. We find it convenient to introduce the following: (Gi) Assume there exists a function a : D3 —» [0, +00) such that P ( y ) - ! [ F ( y ) - F(x) - A(x)(y - x) + F'(y)(z - y)}\\ < fi(i,y, z) (59) for all x,y,z

€ D.

(G2) There exist continuous, nondecreasing functions w, w\, and W2 such that w : D -» [0, +00),

wi, w2 : [0, R] -» [0, +00)

with u;i(0) = 11)2(0) = 0 and a sequence {zn} (n > 0) of points with \\zi\\ < w(zi) < wi(r)

forallz>0

(60)

and for all k € N Jfc+i

fc+i

£ 11*11 <X>(*) 0) is null. 24

(G3) There exist continuous, nondecreasing functions W3, W4 : [0, R] —* [0, +00) with w3(Q) = W4(0) = 0 such that 57 = a(Xi,yi,xi+i)

< w3(r)

for all i > 0

(62)

and for all k € N fc+i

^2cTi<W4{r)

(63)

t=0

for all Xi,yi,Xi+i

€ D and for r € (0,/?].

Let us define the functions tp,0.

(91) (92)

We wish to choose the operators F^ in such a way that a solution x*h of (90) can be found for each h so that lim x*h =x*.

(93)

h—»oo

In practice we will solve the finite dimensional linear equation Ah(xnh)Axl

= -(Fk(xnh)

+ Qh(x*)),

(94)

then set yl = xnh + Axnh,

(95)

and z£+1=x2 + Ax£-z£

(n>0,fc>0).

(96)

We restrict ourselves to a subset £3 C E\ containing elements that have better smoothness properties than the generic elements of E\. That is we assume that {x*,xn,yn,zn,xn

-x*,yn

-x*,Axn}

C E3

(n > 0).

(97)

Let 7T/, : E\ —» E^, h > 0 be a family of linear projection operators. We assume that the family {n^} (li > 0) satisfy a stability condition of the form K ( x ) | | < qh\\x\\,

x 6 E3, qh < q < 00.

(98)

Since ||7rj[(x)|| < 9 h ||7r h (i)|| and -n\ = 717,, we get \\nh(x)\\ < qh\\TTh(x)\\ => qh>l

(h> 0).

(99)

Moreover, we assume that \\x - irh(x)\\ < 6h\\x\\,

xeE3(h>0).

(100)

Using the triangle inequality we can get \\nh(x)\\ < \\x - irh(x)\\ + \\x\\ < 6h\\x\\ + \\x\\ - (1 + Sh)\\x\\. Hence any best possible approximation (98) will imply that qhoc

Thus our discretization method is characterized by the family {Fh,Qh,Ah(-),Trh,6h}

h>0.

(105)

We assume that the domains D^ are such that Uh(irk{x*),p5) CDhCD /i > 0, ps < r*. (106) Furthermore we assume that the discretization is consistent if the conditions (G5) and (G 6 ) that follow are satisfied: (G5) If Uj £ Ei and u e Ei are solutions of the linear equations Ah(nh{x)){uh) Ah(x)(u)

= -(Fh(nh(x)) + Qh(irk(x))), = -(F(x) + Q(x))

(107) (108)

where x € £3 n D, 71^ (1) e D/,, then ||«fc ■■ */,(«)II < eo 0)

(109)

for some positive constant e^ with eg < C2. (Gg) If u^, u^, wf^, w\, wbh € Ej1 and w1, w2,w*,wA,wb tively) solutions of the linear equations

€ £1 are (respec­

MvtiM) = / W i + Kvk - *l)) - Mxlh)}(yl - 4)dt, (no) Jo 2

MV M)

= (Kixl) - Ah(xl))(y2h - x\),

Ak(yl)(w3h) = Qh(yi)~Qh(xl), Ah(yi)(wAk) = (Ah(xl) - Ah(yi))(yi - x\), Ak(y5h)(w5h) = Fh(yl) - Fh(x5h) - Ah(xsh)(yl) 31

(111) (112) - x\)

+

K(yM-Vh),

(113)

MyDiw1) = / [F'{xi + t(yk - xl)) - AixDKyl -- x2h)dt, (114) JQ

A(yl)(w>) =

(F>(xl)-A(xl))(yl-xl),

3

MvDi* ) = Qivl) - Q(xl), A(yti(w4) = (A(xl)-A(y*))(y*-x*h), A(y5h)(w5) = F(y«h) - F(x%) - A(x5h)(yl - x\)

+ Hyl)(zl-vl),

(115) (116) (117) (118)

where the given points ylh, y\, y\, y\, x{, x\, x\, x\, z\ € Dh, then \H"*h{vl)\\<ei6h\\vl-xl\\, \\wl~-Kh{w2)\\<e26h\\yl-xl\\,

(119) (120)

\\v&-*K(v?)\\<e36h\\ti-x\\\,

(121)

iK-TT^^II^e^H^-xtll,

(122)

and \\wl-iTh(wb)\\<es8h

(123)

for some positive constants ei,e2,e3,e4, and esWe find it convenient to define the functions Ch, Cj1, C^, C$, ah by Ch = qhC, C? = qhd, C£ = qhC2, C# = qhC3, ah = e56h + qha

(124)

and the constants 6^,62,63,64 by 6^ = erfh +biqh,b!2l = e2 0) are in U(x0, R) C D, set Dh = Uh(irh(x*),p5). Choose nh(x0) 6 U(x0, f ) , p 5 < f and r* < ^ . Then we can easily see that DhCU(x0,R)

(h>0).

In this case wh functions are just the w functions restricted on D/, (h > 0). Finally we can easily see from (60) and (81) that function tU2 can be identified with the function w$. Choose for example zn = dn for all n > 0, h > 0. However we do not need to use this to prove our discretization results. We can now state the following local results. Theorem 8. Under the hypotheses of Theorem 5, consider a discretization family (105) that satisfies (98), (100), (101) and (106). Moreover, assume: (i) operators F^ are Frechet-differ-entiable, whereas the operators Qh are con­ tinuous on Dh\ (ii) operators F'h(xh) are invertible for each Xh € £>/,; (iii) conditions (G5) and (G 6 ) are satisfied; and (iv) the following are true wh = w, w% = Wk for k = 1,2,3,4,5 and h € (0,/i 4 ]Then there exists an index ho € N and a number r*(h) such that for h> ho and r*(h) € (0,p~,}, equation (90) has a solution x^ satisfying inequality ||*fc-7r,,(z*)|| /14 and r*(h) Q (0,j?5]. We set x\ — 717,(1*) and consider the discretized inexact Newton-like iterations generated by (91)- (92). It is convenient to introduce the approximations u\ = Ah(nh(x'))

-'(^.(^(z")) + gh(^(z*))),

and

*3C0 = \\u'hl Since x* is a solution of Eq. (53), we have u* = A{x-)-l(F{x*)

+ Q{x*)) = 0,

and by using condition (G 5 ) and (109) we get in turn s*o(h) = K l l =■ K

- irh(u')\\ < eoSh < c28h.

(136)

From the analysis in Remark No. 2 there exists an index /14 such that wh = w and tu£ = Wk k = 1,2,3,4,5 on Dh = Uh(iTh(x*),Pi) for all /14 > h. Using (64). (124) and (125) we get iph{r) = ho > h* such that for each h > /15 the sequence of discretized Newton-like iterates (91)-(92) with starting points (147) con­ verges to x^. (b) Moreover, inequalities (142) and (146) are satisfied for h > /i 5 and \Nh - N\ < 1 for

h>h5

(148)

where N,N^ are given by (139) and 140 respectively. Proof. Using Theorem 5 we get that x* €

U(XQ,PI)

l | z o - : r l 0).

Similar remarks can be made for the left-hand sides of conditions (75), (76), and (77). 38

(6) Our results can be further generalized if we can find a continuous, nondecreasing and vanishing at the origin function CQ on [0, R)6 and a null sequence {sn} (n > 0) such that WVn+l - * n + l ! l < Ci(\\Xn

-X0\\,\\yn

-Xoi|,lkn+l

-X0ji,

l k n + 1 - Vn\\, WVn ~ I n I I , P r i l l ) = £n+1 •

It suffices to show that the iteration { 0) is Cauchy etc. A choice for en is given by en =vn

(n > 1)

(see (33) and (43) in [8]).

Many other choices are possible. (7) Theorem 5 can be reduced to Theorem 1 in [14] (see also [18], [19]). Set Q(x) = 0, A(x) = F'(x) {x e D), zn = 0 (n > 0), Ci = C2 = 0 and b2 = b3 = 0. Assume that condition (1.5) in [10] is satisfied, then (74) is also satisfied if we choose 2C3 = C(||x - x 0 ||, ||y - loll, \\y ~ x\\) = ^ l l l / _ x|| and 61 = b*i — 0. Similarly Theorem 5 can be reduced to the corresponding theorems in [14], [18], [19]. Wo also note that the results in Theorem 1.5 in [15] have been obtained under the assumption that the operator F is twice Frechet differentiable, whereas here we only assume that F is only Frechet differentiable. Moreover the values of the crucial "c" constants appearing in (67) and (72) can be found in [8, Remark (8)]. 2.5

Applications

In this section we provide an example where our results apply whereas the ones obtained in [1], [12]-[24] do not. We will use the condition ||^(x 0 )- 1 (F'(x) - F'(y))|| < c||x - y||P

(155)

for some fixed p G (0,1] and all x, y € U(x0, R) C D. For p = 1 condition (155) is called Lipschitz and has been used in [1], [12]-[24], whereas for p e [0,1), (155) is called Holder. In particular the results obtained in [151 cannot be used under Holder continuity assumptions. However ours can. To show that, let us choose A(x) = F'(x), Q{x) = 0 for all x € D, zn = 0 (n > 0), Cx = C 2 = 0, W2 = W4 = 0, 60 = bi = 62 — bi — 64 = 0, and the functions C3, C and Co such that - ^ C 3 ( | | x - 10II, IIJ/ -10II, \\y -x\\)= 1+p

C(\\x - xo||, \\y - x 0 ||, ||y - x||) c (l+p)

39

\\x-y\\p (1-C||X-X0||P)

and Co(||x - xoll) = c\\x - xo|] p

(see also Remark No. 7).

It can then easily be seen that conditions (78) and (79) become c (1 + p ) ( l

•so + 7T-

,y+p
0, |ZJ| > 0, i = 1,2,... ,n - 1 we obtain,

*«

||F'(x) - F'(z)\\ =

{(i+\)h>(*r-,',»)}

^L2

1/2

I

= -h*

max

x/

2

l<J f0

and

R = 2.425626633.

It can easily be seen that with the above values all hypotheses of Theo­ rem 5 are satisfied. Hence Newton's iteration (162) is well defined, remains in U(x0,r0) and converges to a solution x' of Eq. (161) which is unique in U(x0,R). 42

References 1. Allgower, E.L., Bohmer, K., Potra, F.A., and Rheinboldt, W.C. A meshindependence principle for operator equations and their descretizations, SIAM J. Numer. Anal. 23, 1(1986), 160-169. 2. Argyros, I.K. A mesh independence principle for operator equations and their discretizations under mild differentiability conditions, Computing 45 (1990), 265 268. 3. Argyros, I.K. On the convergence of some projection methods with per­ turbation, J. Comput. Appl. Math. 36 (1991), 255-258. 4. Argyros, I.K. Comparing the radii of some balls appearing in connection to three local convergence theorems for Newton's method, Southwest Journal of Pure Appl. Math. 1 (1998), 31-42. 5. Argyros, I.K. On the discretization of Newton-like methods, Internat. J. Computer. Math. vol. 52, (1994), 161-170. 6. Argyros, I.K. A unified approach for constructing fast two-step Newton­ like methods, Monatshefte fur Mathematik 119 (1995), 1-22. 7. Argyros, I.K. On the method of tangent hyperbolas, J. Approx. Appl. 12, 1, (1996), 1-23.

Th.

8. Argyros, I.K. A convergence theorem for Newton-like methods under generalized Chen-Yamamoto type assumptions, Appl. Math. Comp. 61, 1, (1994), 25-37. 9. Argyros, I.K. and Szidarovszky, F. The Theory and Applications of Iter­ ation Methods, C.R.C. Press, Inc., Boca Raton, Florida, U.S.A. (1993). 10. Brown, P.N. A local convergence theory for combined inexact-Newton/finitedifference projection methods, SIAM J. Numer. Anal. 24, 2 (1987), 407-434. 11. Brown, P.N. and Saad, Y. Convergence theory of nonlinear NewtonKrylov algorithms, SIAM J. Optimiz. 4, 2 (1994), 297-230. 12. Chen, X. and Yamamoto, T. Convergence domains of certain iterative methods for solving nonlinear equations, Numer. Fund. Anal. Optimiz. 10 (1 and 2), (1989), 37-48. 43

13. Dembo, R.S., Eisenstat, S.C., and Steihaug, T. Inexact Newton methods, SIAM J. Numer. Anal. 19, 2 (1982), 400-408. 14. Deufihard, P. and Heindl, G., Affine invariant convergence theorems for Newton's method and extensions to related methods, SIAM J. Numer. Anal. Vol. 16, No. 1 (1979), 1-10. 15. Deufihard, P. and Potra, F.A. Asymptotic mesh independence of NewtonGalerkin methods and a refined Mysovskii theorem, SIAM J. Numer. Anal. Vol. 29, No. 5 (1992), 1395-1412. 16. Gutierez, J.M. A new semilocal convergence theorem for Newton's method, J. Comput. Appl. Math. 79 (1997), 131-145. 17. Kantorovich, L.V. and Akilov, E.P. Functional Analysis in Normed Spaces. Pergamon Press, New York (1964). 18. Mysovskii, I. On the convergence of L.V. Kantorovich's method of so­ lution of functional equations and its applications. Dokl. Akad. Nauk SSSR, 70 (1950), 565-568. (In Russian.) 19. Mysovskii, I. On the convergence of Newton's method. Trudy Mat. Inst. Steklov 28 (1949), 145-147. (In Russian.) 20. Potra, F.A. and Ptak, V. Sharp error bounds for Newton's process, Nu­ mer. Math. 34 (1980), 63-72. 21. Potra, F.A. On an iterative algorithm of order 1.839 . . . for solving nonlin­ ear operator equations, Numer. Fund. Anal. Optimiz. 7 (1) (1984-85), 75-106. 22. Potra, F.A. On Q-order and .R-order of convergence, SIAM 'J. Optimiz. Th. Appl. 63, 3 (1989), 415-431. 23. Vainikko, G.M. Galerkin's perturbation method and the general theory of approximate methods for nonlinear equations, USSR Comp. Math. Math. Phys. 4 (1967), 723-751. 24. Zabrejko, P.P. and Nguen, D.F. The majorant method in the theory of Newton-Kantorovich approximations and the Ptak error estimates, Numer. Fund. Anal. Optimiz. 9 (5 and 6), (1987), 671-684.

44

LINEAR INTEGRAL OPERATORS W I T H H O M O G E N E O U S KERNEL: A P P R O X I M A T I O N PROPERTIES IN M O D U L A R SPACES. APPLICATIONS TO MELLIN-TYPE C O N V O L U T I O N O P E R A T O R S A N D TO SOME CLASSES OF F R A C T I O N A L OPERATORS

Carlo Bardaro and Ilaria Mantellini Department of Mathematics University of Perugia, Via Vanvitelli 1, 06123 Perugia, Italy E-mail: [email protected] Contact author: C. Bardaro

Dedicated to the memory of Professor Calogero Vinti, unforgettable Teacher and Man Here we state some modular approximation theorems for a class of linear integral operators with kernels satisfying some general homogeneity assumptions, acting on functions defined over locally compact topological groups. Moreover we study the rates of modular approximation in certain generalized Lipschitz classes, defined by means of a modular functional. Applications to Mellin convolution operators, moment type operators and Erdolyi-Kober fractional operators are given.

1

Introduction

Linear integral operators with homogeneous kernel were considered by several authors in connection with functional inequalities (see e.g. [11], [20], [26], [27], [7], [28]), and Fractional Calculus (see e.g., [19], [24], [32], [21], [7], [28]). Also some particular cases were of interest in Approximation Theory and Calculus of Variation. For example a deep attention was dedicated to the socalled "moment type operators". From the classical work of Professor Calogero Vinti ([33], [34;), many results were obtained in connection with pointwise approximation and convergence properties with respect to variation, lenght and area. We quote here the papers [1], [36], [17], [18], [2], [21], [6], [8]. For an interesting survey on these results we refer to [35]: this is the last paper of Calogero Vinti, written during his illness, and it represents a clear proof of his great will-power and his affection to the mathematical research, which he preserved to the end. Recently in [9] general linear integral operators with homogeneous kernel are studied in connection with modular approximation theory in Musielak-Orlicz spaces. 45

Here we consider integral operators of the form (*)

( T / ) ( 5 ) = / K(s,t)

f(t)

dt,

JG

where G is a locally compact (Hausdorff) topological group, AT is a kernel satisfying a general condition of homogeneity (which seems to be suitable in topological groups, and which was introduced in [7]), and / belongs to the domain of T. We study approximation properties of families of operators like (*) in abstract modular spaces. These general functional spaces were extensively studied by J. Musielak (see [29]) and include the classical Orlicz spaces and the Musielak-Orlicz spaces. In Section 5 we study in particular the order of modular approximation in modular Lipschitz classes lip T (p), where r is a "comparison" function and p is a modular functional. These classes were introduced in [10] and represent modular versions of the classical Zygmund classes in L p -spaces (see [16]). In Section 6 we consider, as a particular case, a family of Mellin-type convolution operators. These operators are recently considered in a series of papers by P.L Butzer and S. Jansche ([12], [13], [14], [15]), in which a very complete theory of Mellin transform and its connection with Fourier analysis and integral operators is developed. In Section 7 we apply the previous theory to the moment operators, thus obtaining new results about the order of approximation in modular Lipschitz classes. Finally in Section 8 we take into consideration a class of fractional integral operators introduced by Erdelyi and Kober (see [19], [24], [32]). 2

Preliminaries

Let G be a locally compact topological group, provided with its Haar measure, denoted by p.. Here, for a sake of simplicity, we suppose G abelian. Let 6 be the neutral element of G and by U we denote a base of measurable neighbourhoods of 6 € G. Let us denote by L°(G) the space of all the measurable functions / : G—* J?, finite a,e. in G. Let p : L°(G)—>RQ — [0, +oo] be a measurable modular functional, that is ^satisfies the following assumptions: (i) p(f) = 0 if and only if / = 0 a.e. (ii) p(-f)

= p(f),

(iii) p(af + 0g) < p(f) + p(g), for every f,g& L°(G), a,(3 > 0, a + (3 = 1, 46

(iv) p{F(t, ■)) is a measurable function of t S G for any globally measurable function F : G x G->R£. By means of the functional p we introduce the vector subspace of L°(G), de­ noted by LP(G), defined by: L»(G) =

{feL°(G):[imp(\f)=0}.

The subspace LP(G) is the modular space generated by p. A general theory of modular spaces can be found in [29] (for special cases see also [25]). The following notions on measurable modulars will be used (see [29], [3], [4]): (a) p is quasi-convex if there is a constant M > 1 such that: P I p(t) h(t, •) dti(t) JG

J

<M I p(t) p(Mh{t, •)) dp{t), JG

l

for every p € L (G), p > 0, with ||p||i = 1 and for any globally measur­ able h : G x G->R£. (b) p is monotone if / , g € L°(G), \f\ < \g\ implies p(f) < p(g). (c) p is finite if \A € LP(G) whenever A is a measurable subset of G such that p,(A) < +00. (d) p is absolutely finite if it is finite and for every e > 0, A > 0, there is a 6 > 0 such that p(Axs) < £, for any measurable subset BcG with /x(B) 0, there is a measurable subset AcG such that p{A) < +oo and p{afxc\A) < e; (e.2) for every e > 0 there is a 6 > 0 such that p(afxB) < £> f° r an Y measurable subset BcG with /x(S) < 6. (f) p is r-bounded if there are a constant C > 1 and a nonnegative measurable function / i : G—>]RQ such that /i(«)—»0, as t—>6, such that: p(f(t + -)) 0. Thus we have also g > 0. Let A > 0 be fixed and let us suppose that the right hand side of (4) is finite. We have: (Tf)(s)-g(s)

[ K(s,t)f{t)dn(t)

- g(s),

JG

for any s e G. Now if s G G is such that (Tf)(s) similar reasonings as in Theorem 1, we obtain: \(Tf)(s)

- g(s) is nonnegative, by

(rHz))- 1 R\g(z + s) - g(s)\ dft{z)

- g(s)\ < f K(0,z) JG

+ \RAK - 1| g(s), while if ,s e G is such that (Tf)(s) have: |(77)(5) - g(s)\ < f K(0,z)

- f(s) is negative, in analogous way we

(v(z))-> r \g(z + s) - g(s)\ dp(z)

JG

+ \rAK-l\

g{s),

and thus, taking into account that r < R, we finally have the following esti­ mation: \{Tf)(s) - g(s)\ < f K(0, z) (viz))'1

R\g(z + s) - g(s)\ dfi(z)

JG

+ &K g{s). From the properties of the modular p we deduce, for A > 0: l -g)]

0: p\\{Tf

- 9)] < p[X(Tf+ - g+)} + p[X(Tf- - g-)},

the inequality (4) follows, by applying the final inequality in the first step to the functions / + and / ~ , and taking into account of the monotonicity of the modular p. □ Remark 2. The previous estimation extend in various directions correspond­ ing results for Orlicz or Musielak-Orlicz spaces (see e.g. [9]). 52

Remark 3. From (4) we deduce that if g = 77/ € LP{G) then Tf-g Examples exist for which / e L"(G) but Tf - g $ L"{G).

e L"(G).

Suppose now 77 e L°°(G) and |J7?||oo > 0. Then for K € JCT), we have:

a(s)=: f K(s,t) dp{t) < AKWVU. JG

We can assume \\r]\\oo = 1- Indeed putting r)*(t) = ^(O/IMIoo.

we

easily have

We have the following Theorem 3. Let 77 e L°°(G) and y ^ = 1, and let K e £ , . f/nder 0, / € LP(G) n DomT, U eU: PlHTf - /)] < Mup(2\MRAKf,

U)

+ ^^-{p{4\MRAKCf) + ^

+

p[A\RMAKf\}

+ p[2X(a(.) - 1) /(•)]

(5)

Proof. The proof is now more direct and we will use only the second inequality in the definition of 77-subhomogeneity of K. As in Theorem 2 we can assume that the right hand side of (5) is finite. Since: \(Tf)(s)

- f(s)\ < / K(s,t)\f(t)

- f(s)\ dfi(t) + \a(s) - 1||/( S )|,

JG

for any s € G, we obtain: ' p [ A ( T / - / ) ] < J + p[2A(Q(.)-l)/(-)] where J=:p

2\ f K(;t)\f{t) -/{•)] d»{t) JG

But for s € G we have:

L

K(s,t)\f(t)-f(s)\dp(t)

< [ K(6,z)

(^z))

~lv(z

+ s)R\f(z

+ s)-

f(s)\dfx(z)

JG


G\U


< M[ A

+ ■¥- [ A

K

o of functions in /C,,. We shall need the constants Aw, Qw, AW(U) introduced is Section 2. Let p be a r-bounded modular on L°(G). If h denotes the function in the definition of r-boundedness of p, we will put: Sw = I (V(z))-1 Kw(8,z)

h{z) dn(z).

JG

We will say that IK is singular if: ( K . l ) s u p U ) > 0 / l U ; = A < +00, l i m u , _ + 0 0 $}„, = 0.

(K.2) For every U £ll we have: lim AKv[U) 54

= 0.

Finally for a singular family IK — (Kw)w>0

we consider the family of operators:

(Twf)(s)-- f Kw(s,t) f(t) dn{t), JG

for any / e X =: f)w>0

DomT,,,.

In this section we will obtain modular convergence theorems for the family {Tw}The following result is an important tool (see [30], [4]). Proposition 1. Let p be a monotone, absolutely finite, absolutely continuous and r-bounded modular on L°(G). Then for every f € LP(G) there is X > 0 such that for every e > 0 there is U eU such that: wP(A/, U) <e,

for 0 < A < A.

By using the previous result we can prove the following theorem. Theorem 4. Let n : G—>iR+ be a measurable function and let p be a mea­ surable, monotone, absolutely finite, absolutely continuous, quasi-convex and r-bounded modular on L°(G). Let IK = (Kw)w>oC)Cv be a singular kernel. Suppose that: lim Sw = 0. (6) Then for every f e X such that y = nf e LP(G) there is A > 0 such that: lim p\\{Twf

- g)\ = 0.

(7)

Proof. Let / € X and g = r\f £ LP(G) and let A > 0 be such that for e > 0 (A5±![/)<e,

for any 0 < A < A

for a suitable U 6 U. Let A > 0 so small such that: AXMCRA < A and p[AX(AR + l)g] < +oo. From (4) applied to Tw, taking into account of the definition of A and of (6), it is sufficient to prove that p[2XQ,wg}—>0 as w—» + oo. Since 2XQW \g\ < 4X(AR + \)\g\, for every w > 0, the assertion follows from the Lebesgue dom­ inate convergence theorem for modular spaces (see [30], Proposition 2). □ 55

Remark 4. By applying Theorem 3 it is possible to give also a modular convergence theorem of Twf towards / when 77 € L°°. Remark 5. All the previous results can be applied to every translationinvariant Kothe function spaces with absolutely continuous and absolutely finite norm. In particular we obtain corresponding results for any LP(G), 1 < p < +00, with Haar measure. 5

Rates of Modular Convergence in Modular Lipschitz Classes

Let T be the class of all the functions r : G-»I?J with r(6) = 0, r(t) ^ 0 for

t±6. Let p be a measurable modular on L°(G). For a fixed r € T, we define the class (see also [10]): liPx(p) = {/ 6 L"(G) :3a>0

with p[a\f(t + ■)- /(•)!] -

0(T(«)),

* - * } , (8)

where, for any two functions f,g € L°(G), f(t) = 0(g(t)), t—>6, means that there is a constant B > 0 and U € U such that | / ( t ) | < B\g(t)\, for t e U. Here, we will assume that the involved kernels are ^-homogeneous func­ tions. The general case is easily obtained with an additional assumption (see Remark 6 below). Let X be the class of all functions £ : MQ—>IRQ such that £(0) = 0 and f(u) > 0, u > 0. For a given measurable function 77 : G—>1R+, let IK = (Kw)w>o be a family of 77-homogeneous functions. Thus we have r = R = 1. Let £ € X be fixed. Then we will say that IK is f-singular if Aw = 1, for every w > 0, and for every U eU AW(U) = 0(£(u>-1)), iu-> + 00. We have the following: Theorem 5. Let £ 6 X and T e T be fixed. Let 77 : G—*IR+ be a measurable function and IK — {Kw)w>oCftn be a ^-singular kernel. Let p be a measurable, monotone, quasi-convex and strongly r-bounded modular on L°(G). Assume that there is U eU such that: f Kw(6, z) (viz))'1 Ju

T(Z) dz = 0((,{w~1)),

w-> + 00.

(9)

If f e LP(G) is such that g rjf € lipT(p), then for sufficiently small A > 0 we have: P[A(T1U/ - g)} = 0(S(w-1)), w^ + 00. (10)

56

Proof. Taking into account that; Aw = 1, for every w > 0, and r = R = 1, we have for A > 0 and U eU: p\HTwf

- g)}

< M f Kw(0, z) (V(z))-1 Ju + M I

Kw(0,z)

p[XRM(g(z + •) -