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Mathematical Analysis and its Applications. Proceedings of the International Conference on Mathematical Analysis and its Applications, Kuwait, 1985 PDF

423 Pages·1988·28.95 MB·English
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Other Titles in the Series PARKER Energy Conservation Measures ASHWORTH Corrosion: Industrial Problems, Treatment and Control Techniques Pergamon Titles of Related Interest AN AND Introduction to Control Systems, 2nd Edition DE LA PUENTE Components, Instruments and Techniques for Low Cost Automation and Applications JAMSHIDI Linear Control Systems KANTOROVICH Functional Analysis MARCHUK Differential Equations and Numerical Mathematics PUGACHEV Probability Theory and Mathematical Statistics for Engineers SANCHEZ Approximate Reasoning in Intelligent Systems Pergamon Related Journals (sample copy gladly sent on request) Analysis Mathematica Bulletin of Mathematical Biology Computers & Mathematics with Applications International Journal of Applied Engineering Education Journal of Applied Mathematics and Mechanics Mathematical Modelling Mathematical Analysis and its Applications Proceedings of the International Conference on Mathematical Analysis and its Applications, Kuwait, 1985 Edited by S. M. MAZHAR, A. HAMOUI and N. S. FAOUR Kuwait University, P.O. Box 5969 Safat, Kuwait PERGAMON PRESS OXFORD · NEW YORK · BEIJING FRANKFURT SAO PAULO SYDNEY TOKYO TORONTO U.K. Pergamon Press pic, Headington Hill Hall, Oxford OX3 OBW, England U.S.A. Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523, U.S.A. PEOPLE'S REPUBLIC Pergamon Press, Room 4037, Qianmen Hotel, Beijing, OF CHINA People's Republic of China FEDERAL REPUBLIC Pergamon Press, Hammerweg 6, OF GERMANY D-6242 Kronberg, Federal Republic of Germany Pergamon Editora, Rua Eca de Queircs, 346, BRAZIL CEP 04011, Paraiso, Säo Paulo, Brazil Pergamon Press Australia, P.O. Box 544, AUSTRALIA Potts Point, N.S.W. 2011, Australia Pergamon Press, 8th Floor, Matsuoka Central Building, JAPAN 1-7-1 Nishishinjuku, Shinjuku-ku, Tokyo 160, Japan Pergamon Press Canada, Suite No. 271, CANADA 253 College Street, Toronto, Ontario, Canada M5T 1R5 Copyright (C) 1988 Pergamon Press pic. All Rights Reserved. No part of this publication may be repro­ duced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers. First edition 1988 Library of Congress Cataloging in Publication Data International Conference on Mathematical Analysis and its Applications (1985 : Kuwait, Kuwait) Mathematical analysis and its applications. (KFAS proceedings series ; v. 3) 1. Mathematical analysis—Congresses. I. Mazhar, S. M. II. Hamoui, A. (Adnan) III. Faour, N. S. (Nazih S.) IV. Title. V. Series. QA299.6.1565 1985 515 88-4055 British Library Cataloguing in Publication Data International Conference on Mathematical Analysis and its Applications (1985 : Kuwait) Mathematical analysis and its applications. 1. Calculus I. Title II. Mazhar, S. M. III. Hamoui, A. IV. Faour, N. S. V. Series 515 ISBN 0-08-031636-0 In order to make this volume available as economically and as rapidly as possible the author's typescript has been reproduced in its original form. This method unfortunately has its typo­ graphical limitations but it is hoped that they in no way distract the reader. Printed in Great Britain by A. Wheaton & Co. Ltd., Exeter ORGANIZING COMMITTEE Professor Fuad S Mulla Professor S M Mazhar Dr Adnan Hamoui* Dr Waleed Deeb Dr Nazih S Faour Dr R Younis Dr M N Al-Tarazi Dr M Al-Zanaidi * Also representative of the Kuwait Foundation for the Advancement of Sciences EDITORIAL COMMITTEE Professor S M Mazhar Dr Adnan Hamoui, Dr Nazih S Faour ORGANIZING COMMITTEE Professor Fuad S Mulla Professor S M Mazhar Dr Adnan Hamoui* Dr Waleed Deeb Dr Nazih S Faour Dr R Younis Dr M N Al-Tarazi Dr M Al-Zanaidi * Also representative of the Kuwait Foundation for the Advancement of Sciences EDITORIAL COMMITTEE Professor S M Mazhar Dr Adnan Hamoui, Dr Nazih S Faour PREFACE An international conference on "Mathematical Analysis and its Applications", sponsored jointly by Kuwait University and the Kuwait Foundation for the Advancement of Sciences, was held in Kuwait from February 18 to February 21, 1985. It was attended by a large number of mathematicians from all over the world. Twenty one invited talks were delivered by eminent mathematicians and about 53 research papers were presented in the form of short communications. The present volume contains the texts of some of the invited talks and research papers presented at the Conference. All articles appearing in these proceedings were duly refereed with the exception of invited talks. It is a pleasure to express, on behalf of the Department of Mathematics, sincere thanks to Kuwait university and the Kuwait Foundation for the Advancement of Sciences for providing necessary funds towards holding the conference and the publication of its Proceedings. We acknowledge our indebtedness to Dr Bader Al-Saqabi, ex-Chairman of the Department of Mathematics, for placing resources of the Department at our disposal. We also express our gratitude to the invited speakers for giving stimulating talks at the conference, and to all participants for making the conference a success. Finally the staff of Pergamon Press merits our warm thanks for publishing the proceedings. S M Mazhar Adnan Hamoui Nazih S Faour 1 May 1987 vii AN ESTIMATE FOR THE RATE OF CONVERGENCE OF A GENERAL CLASS OF ORTHOGONAL POLYNOMIAL EXPANSIONS OF FUNCTIONS OF BOUNDED VARIATIONt R. BOJANIC* Department of Mathematics, Ohio State University, Columbus, OH 43210, USA 1. Let {p (x)} be a sequence of orthogonal polynomials generated by a non-negative weight function ω(χ) on [-1 , 1 ] and 1 h = [ ω(*)ρ*(£) dt. (1.1) n -1 We denote by %(f) = (]/hn) \ "it) f(t) p (t) at , n -1 tt= 0,1,2,... , the Fourier coefficients of / with respect to {p (x)} and by lak(f)pk(x) (k> 0) , the Fourier expansion of /. The n-th partial sum of this series will be denoted by η S (f,x) = J a (f)p (x). n k k k=o In 1952, G. Freud proved in [11] that under certain hypotheses about the weight function ω(χ) it is possible to conclude that lim Sn(f9x) = f{x) (1.2) ft-*oo Invited talk. The author is grateful to the Kuwait University and the Kuwait Foundation for the Advancement of Sciences for the support of his participation at the Conference on Mathematical Analysis and its Applications. 1 2 R. Bojanic if / is a function of bounded variation on [-1,1], continuous at the point #€(-1,1). Freud1 s result in its simplest form can be stated as follows. Suppose that the weight function ω(α?) satisfies the following condi­ tions on (-1,1): (i) ω(χ) < M{\-x2)~A , (ii) \p {x)\ <M{\-x2)~BhU2 . Suppose further that / is a function of bounded variation on [-1,1]. Then (1.2) holds at every point x€ (-1,1) where / is continuous. This result is clearly the analog of the well-known Dirichlet-Jordan test for the convergence of ordinary Fourier series of 2π-periodic functions of bounded variation. We shall consider here the problem of estimating the rate of conver­ gence of the sequence of partial sums {s (f *x)} when / is a function of bounded variation on [-1,1], not necessarily continuous at the point x € (-1,1). In order to obtain the rate of convergence we have to assume that, in addition to (i) and (ii) , the sequence of polynomials {p (x)} satisfies the condition i f i chU1 (iii) J u{t) {t)dt\ <-£- . Vn -1 Our main result can be stated as follows. THEOREM 1. Let {p (#)} be a sequence of orthogonal polynomials on [-1,1] generated by a non-negative weight function (ύ(χ) and let h be defined by (1.1). We assume that for every χ € (-1,1) and n — 1,2,... conditions (i), (ii) and (iii) are satisfied. If / is a function of bounded variation on [-1,1] then n K(f,*)-t(/U 0) (,-0))|<££> 1 V^Y^'ligJ + +/ + \ |/(* + 0) -f(x-O) | \S{$,x)\. (1.3) n x Here, g is defined by 'fit)-f(x-o) if -i<t<x<i gW = { 0 if t = x (1.4) x f{t)-f(x + 0) if -\<x<t<\ and ψ is the special function ». X -1 if t<x ty{t) = sign(t-x) = \ -0 if t=x x (K5) 1 if t > x . Orthogonal Polynomial Expansions of Bounded Variation 3 Also, C(x) > 0 for x £ (-1,1) and V (f) is the total variation of the func­ tion / on [a,b ] . If f is, in addition, continuous at the point x € (-1,1), inequality (1.3) becomes \S (f,x)-f(x) <— I V . (f) · n M+ w n n u — ( i +) j\ k=\ x x Theorem 1 shows that the convergence of the orthogonal expansion of a function of bounded variation, at a point of discontinuity, depends essen­ tially on the behaviour of the orthogonal expansion of the special function ψ (t) = sign(t-x) at the point t=x . It is easy to see that condition (iii) is satisfied if and only if the Fourier coefficients of ψ with respect to {pn(x)} satisfy the inequality n h n It is interesting to observe here that this condition, or condition (iii), which we had to add to Freud's conditions (i) and (ii) in order to obtain an estimate of the rate of convergence, is itself related to the 2 rate of best L -approximation of the function ψ by polynomials. Using orthogonality property of the sequence {p (x)} we have, for any polynomial Q of degree <n- 1 , n 1 W * « ' s ίχ ω(*) **(*>?«(*><** -1 f ω(*)(φ (ί)-β _ (*))ρ2(ί)4ί . Λ κ ι -1 Hence \l/2 / 1 vl/2 Καη{^χ) < (J ω(*) (**(*)-Vi(t)) dt) (l ^Pn{t)dt) -1 -1 Taking infimum over all polynomials of degree <n- \ we find that Κ\αη(*χ)\ <Α^2£^2)(ω.« ), Γ (2) where En (ω»Ψχ) is the constant of best L2-approximation of ψ by poly­ nomials of degree <n-l. Thus, condition (iii) will be satisfied if £un( 2)(^ω>ψ,χψ' ) ^< -η . Theorem 1 is a very useful tool for the study of convergence properties of specific orthogonal expansions of functions of bounded variation. 4 R. Bojanic A particular and yet sufficiently general system of orthogonal polyno­ mials to which Theorem 1 is applicable is the system of Jacobi polynomials l{Pn ' (x) } generated by the weight function ω(χ) = (1 -xf (1 +x)3 (a ß) fn + a\ and normalized so that P * ( 1 ) = I I . We have here n \ n / (α,β) 2α+3+1 Γ(η+α+1) Γ(η+β+1) Α η 2η+α+3+1 Γ(η+1) Γ(η+α+$+1) and, as it is easy to see, . , (a, 3) a+3 ,, ,v Ί lim nh = 2 . (1.6) n-»<*> In case of Jacobi polynomials, properties (i), (ii) and (iii) can be easily established. An estimate for the rate of convergence of the sequence {Sn(4> >%)} to zero, following a suggestion by R. Askey, can be obtained as follows. Since x is an interior point of [-1,1], we can first use the equiconvergence theorem for Jacobi series (see [12], Th.9.12, p.244) to conclude that lim Sn^x>x) = 5 Vx (χ+ °) + Ψ (*-°))= ° > n ->°° χ which is equivalent to the statement that oo It follows then that oo Sn^x>*)=- l αα,, ί(ψψ )) PP,^('pa' 'e) (x) and an estimate for S (ψ , χ) can be obtained by using asymptotic properties of Jacobi polynomials. We have the following estimate of the rate of convergence of the Fourier-Jacobi expansion of a function of bounded variation. THEOREM 2. Let Ϊ? ' (#)f be the sequence of Jacobi polynomials and let / be a function of bounded variation on [-1,1]. I_f_ a >- ^ anc* 3 > ~ 2» we have I C(a,3;a0 V x + (\-x)Ik , ?„(a,3)(f,Ä) -|(f(x + 0) +f(x-0)) v ^ ) γ (q) k=i M(a,$;x) , + ]f(x + 0) -f(x-O)

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