Springer Series in Computational Mathematics 19 Editorial Board R.L. Graham, Murray Hill (NJ) J. Stoer, Warzburg R. Varga, Kent (Ohio) A.A. Gonchar E.B. Saff Editors Progress in Approximation Theory An International Perspective Springer-Verlag New York Berlin Heidelberg London Paris Toyko Hong Kong Barcelona Budapest AA Gonchar E.B. Saff Steklov Mathematics Institute Institute for Constructive Mathematics 117966 Moscow GSP-1 University of South Florida Russia Tampa, FL 33620 USA Mathematics Subject Classification (1991): 30-06, 33-06, 41-06 With 9 figures. Library of Congress Cataloging-in-Publication Data Progress in approximation theory: an international perspectivel edited by A.A. Gonchar and E.B. Saff. p. cm. - (Springer series in computational mathematics; 19.) "Proceedings of an international conference on approximation theory that was held March 19-22, 1990, at the University of South Florida, Tampa"-Pref. Includes bibliographical references (p. 33-35) and index. ISBN-13:978-1-4612-7737-8 1. Approximation theory-Congresses. I. Gonchar, A.A. (Andrei A.) II. Saff, E.B., 1944- . III. Series. QA221.P78 1992 511 '.4-dc20 92-24316 Printed on acid-free paper. © 1992 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1992 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or here- after developed is forbidden. 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Camera-ready copy prepared at the Institute for Constructive Mathematics. 987654321 ISBN-13:978-1-4612-7737-8 e- ISBN-13:978-1-4612-2966-7 DOl: 10.1007/978-1-4612-2966-7 Preface This volume is the proceedings of an international conference on approxi- mation theory that was held March 19 to 22, 1990, at the University of South Florida, Thmpa. The conference was a historic event in the sense that it brought together for the first time a large number of approximation theorists from the United States and from Russia, the Ukraine, and several other former Soviet countries. In addition, 69 other delegates from Europe- an, North American, and Asian countries were present. The conference was hosted by the Institute for Constructive Mathemat- ics, and the organizing committee consisted of A.A. Gonchar, E.B. Saff, S. Khrushchev, R.A. DeVore, and P. Nevai. Both plenary and shorter invited research announcements were presented. This proceedings differs in several important respects from typical confer- ence publications. First, all of the contributions that appear were by invita- tion of the editors. Second, much latitude was granted to authors in prepar- ing their manuscripts; the typical page limitations were relaxed in order to encourage contributions that present not only new results but include de- tailed perspectives on the subject area. This volume is designed to give an overview of current research activities in approximation theory and special functions that truly reflects the interna- tional nature of these subject areas. For example, 8 of the 19 chapters are authored or coauthored by mathematicians from former Soviet countries. The contents include a wide range of topics, which we now briefly describe. Of interest to mathematicians in special functions are new results dealing with q-hypergeometric functions, difference hypergeometric functions, and basic hypergeometric series with Schur function argument. Several chapters concern the theory of orthogonal polynomials and expansions, including generalizations of Szego type asymptotics and connections with Jacobi ma- trices. The convergence theory for Pade and Hermite-Pade approximants is explored in three chapters in which techniques from potential theory are emphasized. The relatively new topics of wavelets and fractals are featured in chapters dealing with invariant measures and nonlinear approximation. Applications of results concerning approximation by entire functions and the problem of analytic continuation are also included. In addition, gener- alizations of de Brange's inequality for univalent functions are presented from an operator point of view in a quasi-orthogonal Hilbert space setting. v vi Preface Further topics include rearrangements of functions, harmonic analysis, nu- merical estimates for the de Bruijn-Newman constant, and approximation by polynomials with varying weights. It is the sincere hope of the organizers that the conference along with this proceedings will act as a catalyst for future joint research and extensive cooperation among approximation theorists of all nations. The organizers are particularly grateful for the support of the National Science Foundation, the former Soviet Academy of Sciences, and the Uni- versity of South Florida. We are also indebted to the local organizing committee consisting of M.E.H. Ismail, M. Parrott, B. Shekhtman, V. Thtik, and C. Williams for their help in planning activities and hosting our foreign guests. Special notes of appreciation are further extended to Maria Carvalho, who served effi- ciently and energetically as the conference secretary, and to Rafael Munoz, who carefully typed many of the manuscripts. Moscow, Russia A.A. Gonchar Thmpa, Florida E.B. Saf! Contents Preface..................................................... v List of Participants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Difference Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 N.M. Atakishiyev and S.K. Suslov Pad~ Approximants for Some q-Hypergeometric Functions. . . . . . . . . 37 M.E.H. Ismail, R. Pertine, and J. Wimp Summation Theorems for Basic Hypergeometric Series of Schur Function Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 S.C. Milne Orthogonal Polynomials, Recurrences, Jacobi Matrices, and Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 P. Nevai Szego lYPe Asymptotics for Minimal Blaschke Products. . . . . . . . . . . . 105 A.L. Levin and E.B. Sa/I Asymptotics of Hermite-Pad~ Polynomials. . . . . . . . . . . . . . . . . . . . . . . 127 A.I. Aptekarev and H. Stahl On the Rate of Convergence of Pad~ Approximants of Orthogonal Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 A.A. Gonchar, E.A. Rakhmanov, and S.p. Suetin Spurious Poles in Diagonal Rational Approximation. . . . . . . . . . . . . . . 191 D.S. Lubinsky Expansions for Integrals Relative to Invariant Measures Determined by Contractive Affine Maps......................... 215 C.A. Micchelli Approximation of Measures by Fractal Generation Thchniques ...... 241 S.Demko Nonlinear Wavelet Approximation in the Space C (Rd) ••••••••••••• 261 R.A. DeVore, P. Petrushev, and X.M. Yu vii viii Contents Completeness of Systems of Translates and Uniqueness Theorems for Asymptotically Holomorphic Functions . . . . . . . . . . . . . . . . . . . . .. 285 A.A. Borichev Approximation by Entire Functions and Analytic Continuation ..... 295 N. U Arakelyan Quasi-Orthogonal Hilbert Space Decompositions and Estimates of Univalent Functions. II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 315 N.K. Nikolskii and V.I. Vasyunin On the Differential Properties of the Rearrangements of Functions. .. 333 v.l. Kolyada A Class of I.M. Vinogradov's Series and Its Applications in Harmonic Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 353 K.l. Oskolkov A Lower Bound for the de Bruijn-Newman Constant A. II.......... 403 T.S. Norfolk, A. Ruttan, and R.S. Varga On the Denseness of Weighted Incomplete Approximations . . . . . . . .. 419 P. Borwein and E.B. Saff Asymptotics of Weighted Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . .. 431 M. v. Golitschek, G. G. Lorentz, and Y. Makovoz List of Participants Dan Amir, Faculty of Exact Sciences, leI-Aviv University, Ramat Aviv 69978, Israel. G.A. Anastassiou, Department of Mathematical Sciences, Memphis State University, Memphis, TN 38152, USA. Milne Anderson, Department of Mathematics, University College, London WC16 EBT, UK. A.I. Aptekarev, Keldysh Institute of Applied Mathematics, Miusskaya Sq. 4, 125047 Moscow A-47, Russia. Norair U Arakelyan, Institute of Mathematics, Armenian Academy of Sci- ences, M~shal Bagramian Ave. 24 - B, 375019 - Yerevan, Armenia. Remi Arcangeli, Laboratoire de Mathematiques Appliquees, Universite de Pau, Avenue de l'Universite, 64000 Pau, France. Richard A. Askey, Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA. Natig Mamed ogly Atakishiyev, Physics Institute, Narimanov pro 33, Baku 370143, Azerbaijan. R V. Atkinson, Department of Mathematics, University of Toronto, Toron- to, Canada M5S IA1. Bogdan M. Baishanski, Department of Mathematics, Ohio State University, Columbus, OH 43210-1174, USA. George A. Baker, Jr., T-I1, MS-B262, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. Laurent Baratchart, Res. des Hartes, 700 ch. des Combes, 06600 Antibe, France. V.I. Belyi, Institute for Applied Mathematics and Mechanics, Ukrainian Academy of Science, ul. Roze Luxemburg 74, Ukraine. Hubert Berens, Mathematisches Institut, UniversiHit Erlangen - Niirnberg, Bismarckstrasse 1112, D-8520 Erlangen, Germany. ix x Participants Hans-Peter Blatt, Lehrstuhl fUr Mathematik-Angewandte Mathematik, Katholische Univ. Eischstatt, Ostenstrasse 18, D-8078 Eichstatt, Germany. Andre Boivin, Department of Mathematics, University of Western Ontario, London, Canada N6A 5B7. Ranko Bojanic, Department of Mathematics, Ohio State University, Co- lumbus, OH 43210-1174, USA. A.A. Borichev, Steklov Mathematical Institute, St. Petersburg Branch, 27 Fontanka, 191011 St. Petersburg, Russia. Peter Borwein, Department of Mathematics, Dalhousie University, Halifax, Canada B3H 4H8. Dietrich Braess, Mathematisches Institut, Ruhr- Universitat, 4630 Bo- chum, Germany. Bruno Brosowski, Universitat Frankfurt, Fachbereich Mathematik, D-6000 Frankfurt a.M., Germany. M.D. Buhmann, DAMTP, University of Cambridge, Silver Street, Cam- bridge CB3 9EW, UK. Paul L. Butzer, Lehrstuhl A. fUr Mathematik, Thchnishe Hochschule Aa- chen, D-5100 Aachen, Germany. Alfred Cavaretta, Jr., Mathematics Department, Kent State University, Kent, OH 44242, USA. Bruce L. Chalmers, Department of Mathematics, University of California, Riverside, CA 92521, USA. Jairo A. Charris, Department of Mathematics and Statistics, OF. 315, Na- tional University of Colombia, Bogota, Colombia. Weiyu Chen, Department of Mathematics, University of Alberta, Edmon- ton, Canada T6G 2Gl. Sandra Cooper, Department of Pure and Applied Mathematics, Washing- ton State University, Pullman, WA 99163-2930, USA. Carl de Boor, Department of Mathematics, University of Wisconsin, Madi- son, WI 53705, USA. Marcel G. de Bruin, Delft University of Thchnology, Faculty of Thchnical Mathematics and Informatics, 2600 A J Delft, The Netherlands. Jesus S. Dehesa, Departamento de Fisica Moderna, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain. Participants xi Biancamaria della Vecchia, Istituto per Applicazioni della Matematica- CNR, 80131 Napoli, Italy. Stephen Demko, School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA. Baiquio Deng, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA. Catherin Detail/e, Department of Mathematics, Facultes Univ. ND de la Paix, Rempart de la Vierge, 8, B-5000 Namur, Belgium. Ron A. De Yore, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA. z. Ditzian, Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G 1. Andre Draux, Laboratoire d'Analyse Numerique, Universite de Lille, 59655 Villeneuve d'Ascq Cedex, France. Michael Eiermann, Institiit fUr Praktische Mathematik, Universitiit Karls- ruhe, D-7500 Karlsruhe 1, Germany. Tamas Erdelyi, Department of Mathematics, Ohio State University, Colum- bus, OH 43210-1174, USA. Mosche Feder, Department of Mathematics, St. Lawrence University, Can- ton, NY 13617, USA. Bernd Fischer, Institute of Applied Mathematics, University of Hamburg, D-2000 Hamburg 13, Germany. Thomas Fischer, Fachbereich Mathematik, Wolfgang Goethe-Universtitiit, 6000 Frankfurt 1, Germany. Wolfgang H. Fuchs, Department of Mathematics, Cornell University, Itha- ca, NY 14853, USA. Dieter Gaier, Department of Mathematics, University of Giessen, Arndt- strasse 2, 63 Giessen, Germany. Jeffrey Geronimo, School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA. Manfred v. Golitschek, Institiit fUr Angewandte Mathematik, Universitiit Wiirzburg, Am Hubland, 8700 Wiirzburg, Germany. Andrei A. Gonchar, Steklov Mathematics Institute, Vavilova 42, 117966 Moscow GSP-l, Russia.