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 Steklov Mathematics Institute 117966 Moscow GSP-1 Russia

E.B. Saff Institute for Constructive Mathematics University of South Florida 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 here­after developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Permission to photocopy for internal or personal use, or the internal or personal use of specific clients, is granted by Springer-Verlag New York, Inc., for libraries registered with the Copyright Clearance Center (CCC), provided that the base fee of $0.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress St., Salem, MA 01970, USA. Special re­quests should be addressed directly to Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA. ISBN-13:978-1-4612-7737-811992$0.00 + 0.20

Production managed by Francine Sikorski; manufacturing supervised by Vincent Scelta. 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 approxi­mation 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 Europe­an, North American, and Asian countries were present.

The conference was hosted by the Institute for Constructive Mathemat­ics, 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 confer­ence publications. First, all of the contributions that appear were by invita­tion of the editors. Second, much latitude was granted to authors in prepar­ing their manuscripts; the typical page limitations were relaxed in order to encourage contributions that present not only new results but include de­tailed 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 interna­tional 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 ma­trices. 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, gener­alizations 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, nu­merical 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 Uni­versity 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 effi­ciently and energetically as the conference secretary, and to Rafael Munoz, who carefully typed many of the manuscripts.

Moscow, Russia Thmpa, Florida

A.A. Gonchar 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 Sci­ences, 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, Toron­to, 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, Co­lumbus, 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 Bo­chum, Germany.

Bruno Brosowski, Universitat Frankfurt, Fachbereich Mathematik, D-6000 Frankfurt a.M., Germany.

M.D. Buhmann, DAMTP, University of Cambridge, Silver Street, Cam­bridge CB3 9EW, UK.

Paul L. Butzer, Lehrstuhl A. fUr Mathematik, Thchnishe Hochschule Aa­chen, 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, Na­tional University of Colombia, Bogota, Colombia.

Weiyu Chen, Department of Mathematics, University of Alberta, Edmon­ton, Canada T6G 2Gl.

Sandra Cooper, Department of Pure and Applied Mathematics, Washing­ton State University, Pullman, WA 99163-2930, USA.

Carl de Boor, Department of Mathematics, University of Wisconsin, Madi­son, 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 Matematica­CNR, 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 Karls­ruhe, D-7500 Karlsruhe 1, Germany.

Tamas Erdelyi, Department of Mathematics, Ohio State University, Colum­bus, OH 43210-1174, USA.

Mosche Feder, Department of Mathematics, St. Lawrence University, Can­ton, 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, Itha­ca, NY 14853, USA.

Dieter Gaier, Department of Mathematics, University of Giessen, Arndt­strasse 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.

xii Participants

Laura Gori, Dipartimento di Metodi e Modelli, "La Sapienza," via Scarpa 10, 00161 Rome, Italy.

William B. Gragg, Department of Mathematics, Naval Post-Graduate School, Monterey, CA 93943, USA.

P.R. Graves-Morris, Department of Mathematics, University of Bradford, Bradford, West Yorks BD7 lDP, UK.

Matthew He, Department of Mathematics, Science, and Technology, Nova University, Ft. Lauderdale, FL 33314, USA.

Weighu Hong, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.

Gary Howell, Department of Applied Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA.

Chai-Chang Hsiao, Department of Mathematics, University of South Caro­lina, Columbia, SC 29208, USA.

Yingkang Hu, Department of Mathematics and Computer Science, Georgia Southern College, Statesboro, GA 30460-8093, USA.

Mourad E.H. Ismail, Department of Mathematics, University of South Florida, Thmpa, FL 33620, USA.

Marie-Paule Istace, Department of Mathematics, Facultes Universite ND de la Paix, Rempart de la Vierge, 8, B-5000 Namur, Belgium.

K.G. Ivanov, Institute of Mathematics, Bulgarian Academy of Sciences, Sofia 1090, Bulgaria.

W.B. Jones, Department of Mathematics, University of Colorado, Boulder, CO 80309-0426, USA.

Henry Kallioniemi, Hagelvagen 302, S-951 48 Lulea, Sweden.

Valery Kalyagin, Gorky Institut Polytechnic, Minina 24, Russia.

Boris S. Kashin, Steklov Mathematics Institute, Vavilova 42, 117966, Mos­cow GSP-l, Russia.

Sergei V. Khrushchev, Steklov Mathematics Institute, Vavilov 42, 117966, Moscow GSP-l, Russia.

S. Kiss, Department of Mathematics, Ohio State University, Columbus, OH 43210-1174, USA.

Fadimba Kolli, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.

Participants xiii

Victor I. Kolyada, Department of Mathematics, Mathematics Institute, Odessa, Ukraine.

Ibm H. Koornwinder, Centre for Mathematics and Computer Science, 1009 AB Amsterdam, The Netherlands.

Rolitza Kovacheva, Institute of Mathematics, Bulgarian Academy of Sci­ences, Sofia 1090, Bulgaria.

George Kyriazis, Department of Mathematics, University of South Caro­lina, Columbia, SC 29208, USA.

Michael Lachance, Department of Mathematics, University of Michigan­Dearborn, Dearborn, MI 48128, USA.

Andrea Lajorgia, Facolta d'Ingegneria, Monteluco di Roio, 67040-I'Aquila, Italy.

David 1. Leeming, Department of Mathematics and Statistics, University of Victoria, Victoria, Canada V8W 2Y2.

F. David Lesley, Department of Mathematics, San Diego State University, San Diego, CA 92182, USA.

Dany Leviatan, School of Mathematics, Tel Aviv University, 69978 Thl Aviv, Israel.

A.L. Levin, Department of Mathematics, Open University of Israel, Max Rowe Educational Center, 16 Klausner Street, P.O. Box 39328, 61392 Thl­Aviv, Israel.

Wu Li, Department of Mathematics, Pennsylvania State University, Univer­sity Park, PA 16802, USA.

X. Li, Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA.

Glenn M. Lilly, 3482 Landsdowne Drive, Apt. 137, Lexington, KY 40517, USA.

Xiaoyan Liu, Department of Mathematics, University of South Florida, Thmpa, FL 33620, USA.

Maria Laura Lo Cascio, Dipartimento di Metodi e Modelli Matematici per Ie Scienze Appticate, via Scarpa, 10-00161 Rome, Italy.

P.D. Loach, School of Mathematics, University of Bristol, Bristol BS6 6LG, UK.

G. Lopez-Lagomasino, Fac. Mat. y Cib., University Habana, Habana 4, Cuba.

xiv Participants

G. G. Lorentz, Department of Mathematics, University of Texas, Austin, TX 78712, USA.

Rudolf A. Lorentz, Postfach 1240, 5205 St. Augustin 2, Germany.

Lisa (Jacobsen) Lorentzen, Division of Mathematical Sciences, University of Trondheim - NTH, N-7034 Trondheim, Norway.

D.S. Lubinsky, Department of Mathematics, Witwatersrand University, WITS 2050, Republic of South Africa.

Wolfgang Luh, Fachbereich 4/Mathematik, Universitat Trier, Postfach 3825, D-5500 Trier, Germany.

Francisco Marcellan, Departamento de Matematica Aplicada, E. T.S. Inge­nieros Industriales, c/o Jose Gutierrez Abascal2, 28006 Madrid, Spain.

David R. Masson, Department of Mathematics, University of Toronto, Toronto, Canada M5S 1A1.

Giuseppe Mastroianni, I.A.M.-C.N.R., via P. Castellino 111, 80131 Napo­li, Italy.

Syed M. Mazhar, Department of Mathematics, Ohio State University, Co­lumbus, OH 43210-1174, USA.

Scott Metcalf, Wallace 402, Eastern Kentucky University, Richmond, KY 40475-3133, USA.

H.N. Mhaskar, 931 East Lemon Avenue, Glendora, CA 91740-3614, USA.

Charles A. Micchelli, IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA.

Stephen C. Milne, Department of Mathematics, Ohio State University, Co­lumbus, OH 43210-1174, USA.

Ram Mohapatra, Department of Mathematics, University of Central Flori­da, Orlando, FL 32816, USA.

Lee Mong-Shu, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.

J. Myjak, Facolta di Ingegnena, Unive. dell'Aquila, 67100 L'Aquila, Italy.

Paul Nevai, Department of Mathematics, Ohio State University, Columbus, OH 43210-1174, USA.

Nikolai K. Nikolskii, Steklov Mathematical Institute, St. Petersburg Branch, Fontanka 27, St. Petersburg, 191011, Russia.

Participants xv

Olav Njastad, Department of Mathematics, University of 1tondheim, N-7034 1tondheim - NTH, Norway.

Martine Olivi, 1655 Ave. St. Lambert, 06100 Nice, France.

Konstantin I. Oskolkov, Steklov Mathematics Institute, Vavilova 42, 117966 Moscow GSP-l, Russia.

Judith Palagal/o Price, Department of Mathematics, University of Akron, Akron, OH 44325-4002, USA.

K. Pan, Department of Mathematics, University of California, Riverside, CA 92521, USA.

Mary Parrott, Department of Mathematics, University of South Florida, Thmpa, FL 33620, USA.

Pencho P. Petrushev, Institute of Mathematics, Bulgarian Academy of Sci­ences, 1090 Sofia, Bulgaria.

A. Pinkus, Department of Mathematics, Thchnion, Haifa, Israel.

Vasil Popov, Department of Mathematics, Thmple University, Philadelphia, PA 19122, USA.

Marc Prevost, USTL Flandres Antois, Laboratoire d'Analyse Numerique et d'Optimisation, B4t M3, 59655 Villeneuve d'Ascq Cedex, France.

T. Price, Department of Mathematics, University of Akron, Akron, OH 44325, USA.

H. Qiao, Department of Mathematics, University of South Florida, Thmpa, FL 33620, USA.

E.A. Rakhmanov, Steklov Mathematics Institute, Vavilov 42, 117966 Mos­cow GSP-l, Russia.

Lothar Reichel, Department of Mathematics, Kent State University, Kent, OH 44242, USA.

Paolo Emilio Ricci, Dipartimento di Metodi e Modelli Matematici, Univer­sita degli Studi di Roma, 10-00161 Rome, Italy.

Ted J. Rivlin, IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA.

Rene Rodriguez, Department of Mathematics, University of Central Flori­da, Orlando, FL 32816, USA.

Frode Ronning, Department of Mathematics and Statistics, University of 1tondheim, AVH, N-7055 Dragvoll, Norway.

xvi Participants

Andre Ronveaux, Department of Physics, Facultes N.D. de la Paix, B-5000 Namur, Belgium.

David Ross, Embry-Riddle University, Daytona Beach, FL 32114, USA.

John A. Roulier, Department of Computer Science and Engineering, U­ISS, University of Connecticut, Storrs, CT 06269-3155, USA.

Stephan Ruscheweyh, Mathematisches Institut, Universitat Wiirzburg, D-8700 Wiirzburg, Germany.

E.B. Saff, Institute for Constructive Mathematics, Department of Mathe­matics, University of South Florida, Thmpa, FL 33620, USA.

Elisabetta Santi, Dipartimento di Energetica, Universita di l'Aquila, 67040 Roio Poggio -l'Aquila, Italy.

Darrell Schmidt, Department of Mathematical Sciences, Oakland Universi­ty, Rochester, MI 48309-4401, USA.

A. Sharma, Department of Mathematics, University of Alberta, Edmon­ton, Canada T6G 2Gl.

Robert Sharpley, Department of Mathematics, University of South Caro­lina, Columbia, SC 29208, USA.

Boris Shekhtman, Department of Mathematics, University of South Flori­da, Thmpa, FL 33620, USA.

Nikolai Shirokov, LOMI, St. Petersburg 193231, Russia.

Jamil A. Siddiqi, Department of Mathematics and Statistics, Laval Univer­sity, Quebec, Canada GIK 7P4.

Rafat Nabi Siddiqi, Department of Mathematics, Kuwait University, Safat, Kuwait 13060.

Mehrdad Simkani, Department of Mathematics, University of Michigan­Flint, Flint, MI 48502-2186, USA.

Herbert Stahl, TFH/FB2, Luxemburger Str. 10, 0-1000 Berlin 65, Germany.

Sergei P. Suetin, Steklov Mathematics Institute, Vavilova 42, 117966 Mos­cow GSP-l, Russia.

Sergei K. Suslov, Kurchatov Institute of Atomic Energy, Moscow 123182, Russia.

G.D. Taylor, Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA.

Participants xvii

J.P. Thiran, Department of Mathematics, Facultes Universite ND de la Paix, Rempart de la Vierge, 8, B-5000 Namur, Belgium.

Suzanne Thiry, Department of Mathematics, Facultes Universite ND de la Paix, Rempart de la Vierge, 8, B-5000 Namur, Belgium.

M. Vittoria Tirone, Dipartimento Metodi e Modelli Matematici, 16, via Antonio Scarpa, 00161 Napoli, Italy.

Vilmos Totik, Department of Mathematics, University of South Florida, Thmpa, FL 33620, USA.

George Vairaktarakus, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.

Walter van Assche, Department of Mathematics, Katholieke Universiteit, Celestijnenlaan 200 B, B-3030 Leuven, Belgium.

V.C. Varadachari, 651 South McKnight Road, St. Paul, MN 55119, USA.

Richard S. Varga, Institute for Computational Mathematics, Kent State University, Kent, OH 44242, USA.

A.K. Varma, Department of Mathematics, University of Florida, Gaines­ville, FL 32611, USA.

Valeri V. Vavilov, Moscow State University, Department of Mathematics and Mechanics, 119899 Moscow, B-234, Russia.

A.L. Volberg, Laboratory of Mathematical Analysis, V.A.-Styeklov Mathe­matical Institute, Fontanka 27, 191011 St. Petersburg, Russia.

Jiasong Wang, Department of Mathematics, Nanjing University, Nanjing, 210008, China.

Franck Wielonsky, Chemin du Verde, 06570 St. Paul de Vence, France.

Carol Williams, Department of Mathematics, University of South Florida, Thmpa, FL 33620, USA.

Jet Wimp, Department of Mathematics and Computer Science, Drexel Uni­versity, Philadelphia, PA 19104, USA.

Nancy Wyshinski, Department of Mathematics, University of Colorado, Boulder, CO 80309-0426, USA.

Xiang Ming Yu, Department of Mathematics, Southwest Missouri State University, Springfield, MO 65804, USA.

John Zhang, Department of Mathematics, Ohio State University, Colum­bus, OH 43210-1174, USA.

xviii Participants

R. Zhang, Department of Mathematics, University of South Florida, Tam­pa, FL 33620, USA.

Zvi Ziegler, Department of Mathematics, Technion, Haifa 32000, Israel.