Approximation, Complex Analysis, and Potential Theory

274 Pages·2001·7.87 MB·English

by  Norair Arakelian (auth.)

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
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© 2001 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2001
Softcover reprint of the hardcover 1st edition 2001
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Table of Contents
Preface vii
Key to group picture Xl
Participants xiii
Contributors xix
Norair ARAKELIAN
Approximation and value distribution 1
David H. ARMITAGE
Uniform and tangential harmonic approximation 29
Thomas BAGBY, Nelson CASTANEDA
Sobolev spaces and approximation problems for differential operators 73
Andre BOIVIN, Paul M. GAUTHIER
Holomorphic and harmonic approximation on Riemann surfaces 107
Huaihui CHEN
On the Bloch constant 129
David DRASIN
Approximation of subharmonic functions with applications 163
Stephen J. GARDINER
Harmonic approximation and its applications 191
Thomas J. RANSFORD
Jensen measures 221
Arne STRAY
Simultaneous approximation in function spaces 239
Index 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 stu
dents 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 refer
ences, 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 holo
morphic 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 holo
morphic 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 dual
ity, 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 el
liptic 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
vii
viii
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 approxi
mation and extension-type approximation. The first type of approximation seeks to approxi
mate 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 Arake
lian. 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 par
ticular 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

Schillings Daigneault Sabidussi David Michalska Hirnyk Radyna Boivin Lessard Ransford Stray Armitage Ancona Gardiner Bagby Drasin Arakelian Gauthier Bonk
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