Approximation, Complex Analysis, and Potential Theory

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Series II: Mathematics, Physics and Chemistry - Vol. 37

Approximation, Complex Analysis, and Potential Theory

edited by

N. Arakelian Institute of Mathematics, National Academy of Sciences of Armenia, Yerevan, Armenia

and

P.M. Gauthier Departement de Mathematiques et de Statistique, Universite de Montreal, Montreal, Quebec, Canada

Technical Editor:

G. Sabidussi Departement de Mathematiques et de Statistique, Universite de Montreal, Montreal, Quebec, Canada

111...

" Springer-8cience+Business Media, B.v.

Proceedings of the NATO Advanced Study Institute on Modern Methods in Scientific Computing and Applications Montreal, Quebec, Canada 3 t014 July 2000

A C.1. P Catalogue record for this book is available from the Library of Congress.

ISBN 978-1-4020-0029-4 ISBN 978-94-010-0979-9 (eBook) DOI 10.1007/978-94-010-0979-9

Printed an acid-free paper

AII Rights Reserved © 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001 Softcover reprint of the hardcover 1st edition 2001 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

Table of Contents

Preface

Key to group picture

Participants

Contributors

Norair ARAKELIAN Approximation and value distribution

David H. ARMITAGE Uniform and tangential harmonic approximation

Thomas BAGBY, Nelson CASTANEDA Sobolev spaces and approximation problems for differential operators

Andre BOIVIN, Paul M. GAUTHIER Holomorphic and harmonic approximation on Riemann surfaces

Huaihui CHEN On the Bloch constant

David DRASIN Approximation of subharmonic functions with applications

Stephen J. GARDINER Harmonic approximation and its applications

Thomas J. RANSFORD Jensen measures

Arne STRAY Simultaneous approximation in function spaces

Index

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Xl

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xix

1

29

73

107

129

163

191

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239

263

Preface

In the summer of 1967, I moved to Montreal for my first (and only) job with my first (and only) wife and our first child (of six) just on time for the beginning of my first international conference, the yearly Seminaire de matMmatiques superieures, which that year was devoted to complex analysis. It turned out to be one of the best conferences I would ever attend. Since then, perhaps in the hope of recapturing the magic of 1967, I have organized two such SMS at the Universite de Montreal, the first in 1993 and the second in 2000. For the latter, I had the great honour of having Academician Norair Arakelian as co-organizer. These are the proceedings.

At the 1967 SMS, Wolfgang Fuchs gave a series of lectures in which he promulgated the striking new results of Arakelian on complex approximation along with its applications to value distribution theory, previously 'available' only in Russian. The lectures of Fuchs changed the direction of my own research to approximation, which would remain my chief interest till the present day.

The courses given at SMS/NATO ASI-2000 were intended for advanced graduate stu­dents and mathematicians at an early stage of their carreer. However, some more 'mature' mathematicians also attended and found the session quite rewarding. In particular, we were blessed with the active participation of Waficka AI-Katifi and Walter Hayman, both of whom had participated in SMS-1967. The active participation of such senior mathematicians added immensely to the richness of the discussions and their avowed appreciation of the session strongly suggests that these proceedings should be of interest not only to budding analysts but to experts as well. Indeed, the lectures, while providing necessary background and refer­ences, also led to the most recent research in the topics covered.

Naturally, approximation was a major thrust of these courses. My own lectures served as an introduction to holomorphic approximation on Riemann surfaces. In Arakelian's lectures we saw how the theory of holomorphic approximation can be applied to solving problems in the value distribution theory of Rolf Nevanlinna. The lectures by David Armitage and Stephen Gardiner expounded the theory of harmonic approximation and delighted us with remarkable applications.

The study of sub harmonic functions yields important information concerning both holo­morphic functions and harmonic functions. Of course, harmonic functions are subharmonic and a most interesting example of a subharmonic function is log III where I is holomorphic. In fact, the lectures of David Drasin (who also participated in the SMS-1967) show that to a very great extent we can approximate subharmonic functions by functions of type log III where I is holomorphic. While subharmonic functions tell us interesting things about holo­morphic ones, information may flow in the opposite direction as well. Thus, complex analysis and potential theory thrive in symbiosis.

The lectures of Thomas Ransford described an approach to potential theory via dual­ity, the key idea being the notion of Jensen measure, an abstraction of the sub-mean-value property of subharmonic functions.

Besides subharmonic approximation, another way to generalize both holomorphic and harmonic approximation, is to consider various approximation problems associated with el­liptic differential operators. This was the point of view adopted in the lectures by Thomas Bagby. In this context, the natural function spaces to consider are Sobolev spaces.

A given function I may of course belong to various function spaces and the need may

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arise to approximate f simultaneously in more than one of these spaces. Arne Stray's lectures addressed this issue.

Approximation theory can be approached from two viewpoints: smoothing-type approxi­mation and extension-type approximation. The first type of approximation seeks to approxi­mate a given function by a nicer function defined on (more or less) the same domain whereas extension-type approximation seeks to approximate a given function by a function (of the same sort but) defined on a larger domain. All of the approximations in these lectures are of the extension-type with the exception of those discussed by Drasin, which are of smoothing type.

Since Louis de Branges confirmed the Bieberbach conjecture in 1985, the outstanding unanswered question in complex analysis has perhaps been that of determining the precise value of the Bloch constant. In 1990, after more than a half-century without progress, Mario Bonk made a major breakthrough by developing a new distortion theorem which allowed him to improve the known estimate on the Bloch constant. At SMS-2000 Bonk lectured on other (related) matters regarding hyperbolic spaces. Unfortunately, his lectures are not included in these Proceedings. However, the lectures of Chen Huai-hui gave an up-to-the-minute survey on the Bloch constant, including the best estimate at this time, due to Chen and myself.

Another series of lectures which are unfortunately not included in these Proceedings are those of Alano Ancona on the boundary Harnack principle. Subsequently and independently, Paul Koosis also gave a series of lectures in Montreal on the boundary Harnack principle, and produced a set of notes thereof. Those interested in this topic may wish to contact Paul Koosis for a copy of these notes.

I wish to express my sincere thanks to all the lecturers and participants for having helped to make this session such a success and most particularly to my co-organizer, Norair Arake­lian. Special thanks are due to Ghislaine David, coordinator of the SMS, who contributed immeasurably to the preparation, mise-en-scene, and "aftermath". Also, my thanks go to Gert Sabidussi and Arik Loinaz for their excellent work in editing the present volume. The Organizing Committee is most grateful to NATO for financially supporting this session as an Advanced Study Institute. It is a pleasure to proclaim my love and appreciation to my wife, Sandy, for the hospitality she has extended to visitors throughout my carreer and in par­ticular during this AS!. Finally, I extend my congratulations and appreciation to my friend, Aubert Daigneault, director of the SMS for many years, this having been his last. Bravo et merci, Aubert!

This preface began with my quest to recapture the magic of the SMS-1967. For me personally, the magic of SMS-1967 was in large part the work of Arakelian via the lectures of Fuchs. At SMS-1967, Arakelian was present in spirit and Fuchs in the flesh. At SMS-2000 it went the other way. At the excellent suggestion of my co-organizer, Academician Norair Arakelian, we dedicate the present Proceedings to the memory of Wolfgang Fuchs.

Paul M. Gauthier

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Participants

Victor ANANDAM Department of Mathematics King Saud University P.O. Box 2455 11451 Riyadh Saudi Arabia vanandam~ksu.edu.sa

Alano ANCONA MatMmatiques - BiUiment 425 Universite de Paris-Sud 91405 Orsay Cedex France ancona~ath.u-psud.fr

Cristoph BECKER FB IV Mathematik Universitiit Trier 54286 Trier Germany beck4501~uni-trier.de

Charaf BENSOUDA Faculte des sciences Universite Ibn Tofail B.P. 133 Kenitra Morocco

Lucian BEZNEA Inst. Mathematics "Simion Stoilow" Romanian Academy P.O. Box 1-764 70700 Bucharest Romania beznea~stoilow.imar.ro

Leokadia BIALAS-CIEZ Institute of Mathematics Jagellonian University ul. Reymonta 4 30-059 Krakow Poland bialas~im.uj.edu.pl

Bjarte BOE Department of Mathematics University of Bergen Johannes Brunsgate 12 5008 Bergen Norway bjarte.boe~mi.uib.no

Mario BONK Department of Mathematics University of Michigan 525 East University Ave. Ann Arbor, MI 48109-1109 USA m.bonk~math.lsa.umich.edu

Miroslav BRZEZINA Department of Applied Mathematics Technical University of Liberec Halkova 6 461 17 Liberec Czech Republic miroslav.brzezina~vslib.cz

Maria Isabel CAQAO Departamanto de Matematica Universidade de Aveiro Campus Santiago 3810-193 Aveiro Portugal isabelc~at.ua.pt

Yin CHEN Dep. de mathematiques et de statistique Universite Laval Quebec (QC), GlK 7P4 Canada chen~at.ulaval.ca

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Arthur DANIELYAN Department of Mathematics University of South Florida Tampa, FL 33620-5700 USA adaniely~math.usf.edu

Juan Jesus DONAIRE Departament de Matematiques Universitat Autonoma de Barcelona 08193 Bellaterra-Barcelona Spain donaire~mat.uab.es

Alexandre EGOROV Sobolev Institute of Mathematics Pro Akad. Koptyug 4 630090 Novosibirsk Russia yegorov~math.nsc.ru

Juan Carlos FARINA Dpto. Analisis Matematico Universitad de La Laguna C/ Astrofisico Fco. Sanchez 38271 La Laguna, Tenerife Spain jcfarina@ull.es

Richard FOURNIER Dep. de matMmatiques et de statistique Universite de Montreal CP 6128 - Succ. Centre-ville Montreal (QC), H3C 3J7 Canada fournier@dms.umontreal.ca

Sorin GAL Department of Mathematics University of Oradea Str. Armatei Romane 5 37000radea Romania galso@math.uoradea.ro

Nihat G6khan GOGUS Department of Mathematics Middle East Technical University 06531 Ankara Thrkey ggogus~arf.math.metu.edu.tr

Anders GUSTAFSSON Department of Mathematics Umea University 90187 Umea Sweden anders.gustafsson~math.umu.se

Margaret M. HANLEY Department of Mathematics University College Dublin 4 Dublin Ireland mary.hanley@ucd.ie

Waficka HAYMAN c/o Prof. Walter Hayman Department of Mathematics Imperial College SW7 2B2 London United Kingdom

Walter K. HAYMAN Department of Mathematics Imperial College SW7 2B2 London United Kingdom

Markiyan HIRNYK Dept. of Mathematics and Statistics Lviv Academy of Commerce 10, Thhan-Baranovskyi St. 79008 Lviv Ukraine hirnyk~lac.lviv.ua

Baoguo JIANG Dept. Mathematics - Middlesex College University of Western Ontario London (ON), N6A 5B7 Canada bjiang@julian.uwo.ca

H. Turgay KAPTANOGLU Department of Mathematics Middle East Technical University 06531 Ankara Turkey kaptan@arf.math.metu.edu.tr

Katarzyna KIEPIELA Department of Mathematics Technical University Radom J. Malczewskiego St. 29 26-600 Radom Poland

Marta KOSEK Instytut Matematyki Uniwersytet Jagiellonski ul. Reymonta 4 30-059 Krakow Poland marta.kosek@im.uj.edu.pl

Janina KOTUS Department of Mathematics Technical University of Warsaw Plac Politechniki 1 00-661 Warsaw Poland janinak@snowman.impan.gov.pl

Roman LAVICKA Mathematical Institute Charles University Sokolovska 83 186 75 Praha 8 Czech Republic lavicka@karlin.mff.cuni.cz

Norman LEVENBERG Department of Mathematics Rawles Hall Indiana University Bloomington, IN 47405 USA levenberg~scitec.auckland.ac.nz

Jose LLORENTE Departament de Matematiques Universitat Autonoma de Barcelona 08193 Bellaterra-Barcelona Spain gonzalez~math.uab.es

Wolfgang L UH FB IV Mathematik Universitiit Trier 54286 Trier Germany luh~uni-trier.de

Iwona MARCULA Department of Applied Mathematics M. Curie-Sklodowska University Sq. Marie Curie-Sklodowska 5 20-031 Lublin Poland imarcula@ramzes.umcs.lublin.pl

Malgorzata MICHALSKA Institute of Mathematics M. Curie-Sklodowska University Sq. Marie Curie-Sklodowska 1 20-031 Lublin Poland daisy@golem.umcs.lublin.pl

Konstantin OSIPENKO Department of Mathematics MATI - Russian State Tech. University Orshanskaya, 3 121552 Moskow Russia konst@osipenko.mccme.ru

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Pietro POGGI-CORRADINI Department of Mathematics Kansas State University Manhattan, KS 66506 USA pietro~math.ksu.edu

Mohamad R. POURYAYEVALI Department of Mathematics University of Isfahan P.O. Box 81745-163 Isfahan Iran

Igor PRITSKER Department of Mathematics Oklahoma State University Stillwater, OK 74078-1058 USA igor~ath.okstate.edu

Aliaksandr RADYNA Faculty of Mechanics & Mathematics Belarusian State University F. Skaryna Ave. 4 220050 Minsk Belarus radynoa~.bsu.unibel.by

Alexander RASHKOVSKII Institute for Low Temperature Physics 47 Lenin Pr. 61164 Kharkov Ukraine yarchyk~imath.kiev.ua

Dominic ROCHON Dep. de mathematiques et de statistique Universite de Montreal CP 6128 - Suce. Centre-ville Montreal (QC), H3C 3J7 Canada rochon~dms.umontreal.ca

Pamela RYAN Department of Mathematics Univ. of South Carolina, Columbia Columbia, SC 29208-0001 USA pryanOOO~th.sc.edu

Bettina SCHILLINGS FB IV Mathematik Universitat Trier 54286 Trier Germany schi4501~uni-trier.de

Gerald SCHMIEDER FB 6 Mathematik Universitat Oldenburg PF 2503 26111 Oldenburg Germany schmieder~mathematik.uni-oldenburg.de

Haikel SKHIRI Dep. de matMmatiques et de statistique Universite Laval Quebec (QC), GlK 7P4 Canada skhiri~mat.ulaval.ca

Manfred STOLL Department of Mathematics University of South Carolina Columbia, SC 29208 USA stoll~math.sc.edu

Jan SZYNAL Department of Applied Mathematics M. Curie-Sklodowska University Sq. Marie Curie-Sklodowska 5 20-031 Lublin Poland ysszynal~golem.unics.lublin.pl

Serguei VODOP'YANOV Sobolev Institute of Mathematics Akademika Koptyuga Ave. 4 630090 Novosibirsk Russia vodopis~th.nsc.ru

Chung-Chun YANG Department of Mathematics Hong Kong University of Sci. & Tech. Clear Water Bay Kowloon Hong Kong, China mayang~uxmail.ust.hk

Anush YAVRIAN Department of Mathematics Yerevan State University AI. Manoogian Str. 1 375049 Yerevan Armenia

Eduardo S. ZERON Depto. Matematicas CINVESTAV Apdo. Postal 14-740 07000 Mexico D.F. Mexico eszeron~th.cinvestav.mx

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Contributors

Norair ARAKELIAN Institute of Mathematics Armenian National Academy of Sciences Marshall Bagramian Ave. 24b 375 019 Yerevan Armenia arakel~instmath.sci.am

David ARMITAGE Department of Pure Mathematics Queen's University of Belfalst Belfast, BT7 INN Northern Ireland d.armitage~queens-belfast.ac.uk

Thomas BAGBY Department of Mathematics Rawles Hall Indiana University Bloomington, IN 47405 USA bagby~indiana.edu

Andre BOIVIN Department of Mathematics, MC University of Western Ontario London (ON), N6A 5B7 Canada boivin~uwo.ca

Nelson CASTANEDA Department of Mathematical Sciences Central Connecticut State University P.O. Box 4010 New Britain, CT 06050-4010 USA ncastaneda~earthlink.net

Huaihui CHEN Department of Mathematics Nanjing Normal University Nanjing, Jiangsu 210024 China hhchen~publicl.ptt.js.cn

David DRASIN Department of Mathematics Purdue University West Lafayette, IN 47907-1395 USA drasin~math.purdue.edu

Stephen J. GARDINER Department of Mathematics University College Dublin 4 Dublin Ireland stephen.gardiner~cd.ie

Paul M. GAUTHIER Dep. de matMmatiques et de statistique Universite de Montreal CP 6128 - Succ. Centre-ville Montreal (QC), H3C 3J7 Canada gauthier~dms.umontreal.ca

Thomas J. RANSFORD Dep. de mathematiques et de statistique Universite Laval Quebec (QC), GlK 7P4 Canada ransford~math.ulaval.ca

Arne STRAY Department of Mathematics University of Bergen Johannes Brunsgate 12 5008 Bergen Norway stray~mi.uib.no

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