Lecture Notes ni Mathematics Edited by .A Dold and .B Eckmann 5011 Rational Approximation and Interpolation Proceedings of the United Kingdom - United States Conference held at Tampa, Florida, December 12-16, 1983 Edited by .P .R Graves-Morris, .E .B Saff and R.S. Varga lreV-regnirpS Berlin Heidelberg New York Tokyo 1984 Editors Graves-Morris Russell Peter lacitamehtaM of Kent University Institute, ,yrubretnaC Kent 2TC ,FN7 dnalgnE Edward .B Saff for Center lacitamehtaM of University Services, South Florida ,apmaT 33620, Florida USA Richard .S agraV for Computational Institute Mathematics, Kent University State ,tneK Ohio 44242, USA AMS Subject Classification :)0891( 60-14 ISBN 3-540-13899-4 Berlin Heidelberg York Springer-Verlag New oykoT ISBN 0-38?43899-4 Heidelberg Berlin York New Springer-Verlag oykoT This work ts subject to copyright. lLA rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 36 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210 [ (] ) Preface This volume contains the proceedings of the Conference on Rational Approximation and Interpolation, which took place December 12-16, 1983 at the University of South Florida, Tampa, Florida. The conference was held under the auspices of the U.K. - U.S. Cooperative Science Program, an informal agreement between the U.S. National Science Foundation and the U.K. Science and Engineering Research Council to promote and support mutually beneficial scientific activities. The primary purpose of the conference was to bring together pure and applied mathematicians, physicists and engineers to exchange information and set objectives for future research efforts dealing with rational approximation and interpolation. P.R. Graves-Morris and E.B. Saff were the primary organizers of the conference. There were 28 participants from the U.S., 14 from the U.K. and 14 others representing ii additional countries. The number of conference members was kept limited so as to promote dis- cussion among members with diverse backgrounds, in accordance with the aims of the U.K. - U.S. Cooperative Science Program. The contributions to this volume include original research papers as well as a few survey articles. All of these papers were refereed and we are grateful to many advisors for their diligence. It is hoped that this volume reflects the breadth of interest in rational approximation and interpolation, and serves as a source of inspiration for further research. We wish to thank the U.S. National Science Foundation and the U.K. Science and Engineering Research Council for sponsoring the participants from their respective countries. We are also indebted to the University of South Florida (USF) Division of Sponsored Research for the support of the other conference partcipants. The conference planning and activities were facilitated by the USF Center for Mathematical Services, the USF organizing committee consisting of Prof. M. Blake and Prof. J. Snader, and the conference co-host, VI Prof. R.S. Varga. The efforts of these individuals far exceeded the norm. The secretarial help provided by Mary Baroli is also deserving of accolade as were the efforts of several staff members and students in the USF Department of Mathematics. We are further indebted to several companies in the Tampa area for having provided additional support for the conference functions. P.R.G.-M., E.B.S., R.S.V. LIST OF CONTRIBUTORS AND PARTICIPANTS .J MILNE ANDERSON, Mathematics Department, University College, London WCIE6BT, United Kingdom CHRISTOPHER .T .H BAKER, Department of Mathematics, The University, Manchester MI3 9PL, United Kingdom GEORGE A. BAKER, JR., Theoretical Division, Los Alamos National Laboratory, University of California, Los Alamos, New Mexico 87545 MICHAEL BARNSLEY, School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332 AI-PING BIEN, Institute of Applied Mathematics, National Cheng- Kung University, Tainan, Taiwan 700, Republic of China H.-P. BLATT, Katholische Universitat Eichstatt, Mathematish- Geographische Fakultat, Ostenstrasse 26-28, 8078 Eichst~tt, West Germany N. K. BOSE, Departments of Electrical Engineering and Mathematics, 348 Benedum Hall, University of Pittsburgh, Pittsburgh, Pennsylvania 15621 DIETRICH BRAESS, Institut fur Mathematik, Ruhr-Universitat, D-4630 Bochum, West Germany CLAUDE BREZINSKI, Laboratoire d'Analyse Num~rique et d'Optimisation, Universit~ de Lille ,I 59655-Villeneuve d'Ascq Cedex, France MARCEL .G de BRUIN, Department of Mathematics, University of Amsterdam, Roetersstraat 15, 1018 WB Amsterdam, The Netherlands A. BULTHEEL, Department of Computer Science, .K U. Leuven, Celestijnenlaan 200A, B-3030 Leuven, Belgium AMOS .J CARPENTER, Department of Mathematical Sciences, Butler University, 4600 Sunset Avenue, Indianapolis, Indiana 46208 FUHUA CHENG, Institute of Computer and Decision Sciences, National Tsing Hua University, Hsinchu, Taiwan 300, Republic of China .J .S R. CHISHOLM, Mathematical Institute, University of Kent, Canterbury, Kent, CT2 7NF, England CHARLES K. CHUI, Department of Mathematics, Texas A & M University, College Station, Texas 77843 .J .G CLUNIE, Mathematics Department, Open University, Milton Keynes MK7 6AA, United Kingdom lV STEPHEN .G DEMKO, School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332 ALBERT EDREI, Department of Mathematics, Syracuse University, Syracuse, New York 13210 .S .W ELLACOTT, Department of Mathematics, Brighton Polytechnic, Brighton BN2 4GJ, England MICHAEL .E FISHER, Baker Laboratory, Cornell University, Ithaca, New York 14853 W. .H FUCHS, Mathematics Department, Cornell University, Ithaca, New York 14853 J. °L GAMMEL, Department of Physics, Saint Louis University, St. Louis, Missouri 63103 .I GLADWELL, Department of Mathematics, University of Manchester, Manchester MI3 9PL, United Kingdom W° B. GRAGG, Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506 .P .R GRAVES-MORRIS, Mathematical Institute, University of Kent, Canterbury, Kent CT2 7NF, England JOHN .A GREGORY, Department of Mathematics and Statistics, Brunel University, Uxbridge UB8 3PH, England M. GUTKNECHT, Eidgenossiche Technische Hochschule, Seminar fur Angewandte Math., ETH-Zentrum HG, ZUrich CH-8092, Switzerland M. HASSON, Department of Mathematics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213 .T H~VIE, UNIT/NTH, Alfred Getz vei ,i N7034 Trondheim, Norway JEANETTE VAN ISEGHEM, Laboratoire d'Analyse Num~rique et d'Optimisation, Universit~ de Lille ,I 59655-Villeneuve d'Ascq Cedex, France ARIEH ISERLES, King's College, University of Cambridge, Cambridge CB4 ILE, England LISA JACOBSEN, Matematisk Institutt, NLHT Trondheim, Norway WILLIAM B. JONES, Department of Mathematics, University of Colorado, Boulder, Colorado 80309 THOMAS KOVARI, Department of Mathematics, Imperial College, 180 Queen's Gate, London SW7 2BZ, England MICHAEL .A LACHANCE, Department of Mathematics, University of Michigan, Dearborn, Michigan 48128 VII G. L6PEZ LAGOMASINO, Faculty of Physics and Mathematics, University of Havana, Havana, Cuba D. LEVIATAN, School of Mathematics, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel G. .G LORENTZ, Department of Mathematics, University of Texas, Austin, Texas 78712 R. .A LORENTZ, GMD, Schloss Birlinghoven, 5205 St. Augustin ,i West Germany ARNE MAGNUS, Department of Mathematics, Colorado State University, Ft. Collins, Colorado 80523 J. C. MASON, Department of Mathematics and Ballistics, Royal Military College of Science, Shrivenham, Swindon, Wiltshire, England ATTILA MATE, Department of Mathematics, Brooklyn College of the City University of New York, Brooklyn, New York 11210 C° A° MICCHELLI, Mathematical Sciences Department, IBM Thomas .J Watson Research Center, Yorktown Heights, New York 10598 .H N. MHASKAR, Department of Mathematics, California State University, Los Angeles, California 90032 B. NELSON, Department of Mathematics, University of Manchester, Manchester MI3 9PL, England PAUL NEVAI, Department of Mathematics, The Ohio State University, Columbus, Ohio 43210 SYVERT .P N~RSETT, Mathematics Department, university of Trondheim, Trondheim, Norway. Visiting Professor, Department of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3GI, Canada J. NUTTALL, Department of Physics, University of Western Ontario, London, Ontario N6A 3K7, Canada M. .J D. POWELL, DAMTP, Silver Street, Cambridge CB3 9EW, England D. C. POWER, McDonnell Aircraft Company, McDonnell Douglas Corporation, St. Louis, Missouri 63166 T. RIVLIN, Mathematical Sciences Department, IBM Thomas .J Watson Research Center, Yorktown Heights, New York 10598 D. .E ROBERTS, Mathematics Department, Napier College, Colinton Rd., Edinburgh, Scotland S. RUSCHEWEYH, Mathematisches Institut, Universitat W~rzburg, D- 8700 Wurzburg Am Hubland, West Germany llJV A. RUTTAN, Department of Mathematics, Kent State University, Kent, Ohio 44242 E. B. SAFF, Department of Mathematics, University of South Florida, Tampa, Florida 33620 A. SHARMA, Mathematics Department, University of Alberta, Edmonton T6G 2GI, Canada XIE-CHANG SHEN, Department of Mathematics, Peking University, Beijing, China A. D. SNIDER, Department of Electrical Engineering, University of South Florida, Tampa, FLorida 33620 ALLAN STEINHARDT, School of Electrical and Computer Engineering, Oklahoma State University, Stillwater, Oklahoma 74078 F. STENGER, Department of Mathematics, University of Utah, Salt Lake City, Utah 84112 D. F. STYER, Hill Center for the Mathematical Sciences, Busch Campus, Rutgers University, New Brunswick, New Jersey 08903 R° M. THOMAS, Department of Mathematics, U.M.I.S.T., Manchester M60 IQD, United Kingdom W. J. THRON, Department of Mathematics, Campus Box 426, University of Colorado, Boulder, Colorado 80309 VILMOS TOTIK, Bolyai Institute, University of Szeged, 6720 Szeged, Hungary LLOYD N. TREFETHEN, Courant Institute of Mathematical Sciences, New York University, New York, New York 10012 STEWART R. TRICKETT, Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3GI, Canada G. TROJAN, Department of Physics, University of Western Ontario, London 72, Ontario N6A 3K7, Canada JOSEPH L. ULLMAN, Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1003 R. S. VARGA, Department of Mathematics, Kent State University, Kent, Ohio 44242 V. V. VAVILOV, Faculty of Mechanical and Mathematics, University of Moscow, Moscow, U.S.S.R. HANS WALLIN A Department of Mathematics, University of Ume~, S-901 87 UmeM, Sweden G. A. WATSON, Department of Mathematical Sciences, University of Dundee, Dundee DDI 4HN, Scotland J. WILLIAMS, Department of Mathematics, University of Manchester, Manchester MI3 9PL, England CONTENTS Preface III V List of Contributors and Participants SURVEY ARTICLES The Faber Operator ............................................. 1 J. Milne Anderson Survey on Recent Advances in Inverse Problems of Pad~ Approximation Theory ........................................... ii Go L~pez Lagomasino and V° V. Vavilov Some Properties and Applications of Chebyshev Polynomial and Rational Approximation ......................................... 27 J. C. Mason Polynomial, Sinc and Rational Function Methods for Approximating Analytic Functions ............................... 49 F. Stenger APPROXIMATION AND INTERPOLATION THEORY Rational Approximation of Fractals ............................. 73 M. F. Barnsley and S. G. Demko On Rational Approximation of the Exponential and the Square Root Function .................................................. 89 D. Braess Pad~-Type Approximants and Linear Functional Transformations ... 100 C. Brezinski and J. van Iseghem Continued Fraction Solution of the General Riccati Equation .... 109 J. S. R. Chisholm Order Stars, Contractivity and a Pick-Type Theorem ............. 117 A. Iserles Bernstein and Markov Inequalities for Constrained Polynomials .................................................... 125 M. A. Lachance Multivariate Interpolation ..................................... 136 G. G. Lorentz and R. A. Lorentz The Strong Uniqueness Constant in Complex Approximation ........ 145 T. Rivlin On the Minimum Moduli of Normalized Polynomials ................ 150 S. Ruscheweyh and R. S. Varga BLOCK STRUCTURE On the Block Structure of the Laurent-Pad~ Table ............... 160 A. Bultheel Square Blocks and Equioscillation in the Pad~, Walsh, and CF Tables ......................................................... 170 L. N. Trefethen CIRCUIT THEORY Properties of Pad~ Approximants to Stieltjes Series and Systems Theory ................................................. 182 N. K. Bose Degree of Rational Approximation in Digital Filter Realization .................................................... 189 C. K. Chui and X. C. Shen Applications of Schur Fractions to Digital Filtering and Signal Processing .............................................. 210 W. B. Jones and A. Steinhardt CONVERGENCE THEORY A de Montessus Theorem for Vector Valued Rational • nterpolants ................................................... 227 P. R. Graves-Morris and E. B. Saff On the Convergence of Limit Periodic Continued Fractions K(an/l), where a n~ -1/4 ............ ~ ......................... 243 L. Jacobsen ana A. Magnus On the Uniform Approximation of Holomorphic Functions on Convex Sets by Means of Interpolation Polynomials .............. 249 T. K6vari On Equiconvergence of Certain Sequences of Rational Interpolants .................................................. 256 E. B. Saff and A. Sharma Convergence and Divergence of Multipoint Pad~ Approximants of Meromorphic Functions ....................................... 272 H. Wallin CRITICAL PHENOMENA Approximate Analytic Continuation Beyond the First Riemann Sheet .......................................................... 285 G. A. Baker, Jr. Critical Exponents for the General Spin Ising Model Using the Rational Approximation Method .............................. 295 J. L. Gammel, J. Nuttall and D. C. Power