Analytic Capacity and Rational Approximation
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50 Lawrence Zalcman Massachusetts Institute of Technology, Cambridge
Analytic Capacity and Rational Approximation 1968
Springer-Verlag· Berlin· Heidelberg· New York
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. © by Springer-Verlag Berlin' Heidelberg 1968 Library of Congress Catalog Card Number 68-19414. Printed in Germany. Title No. 7370
PREFACE
The purpose of these notes is to make available in English a reasonably complete discussion of some recent results in rational approximation theory obtained by Soviet mathematicians.
More specifically, we shall be concerned
with recent theorems of Vitushkin and Melnikov concerning (qualitative) approximation by rational functions on compact sets in the plane.
Accordingly, we shall have nothing to
say about problems of best approximation, or of approximation on "large" planar sets, or of approximation in the space of n
complex variables
(n > 1).
Each of these subjects is an
active discipline in its own right and deserves its own (separate) treatment.
On the other hand, since problems of
rational approximation have a "local" character, our theorems are related to questions of approximation on regions on Riemann surfaces; however, we shall not pursue that line of thought to any extent. Since our principal desire is exposition, we have tried to keep the prerequisites a minimum.
for understanding the material at
A knowledge of basic function theory and functional
analysis plus a willingness to pursue a few references given in the text are all that is required.
On occasion, it has
been convenient to suppress the details of a proof in the interests of exposition; in each such case the reader w'ill supply the missing steps easily.
In general, however, when
a point has seemed to me obscure, I have chosen to say more rather than less by way of explication. Although this is not primarily a research paper, it does contain some new' material:
a few of the examples and the
unacknowledged contents of sections 7 and 8 have not appeared in print previously.
There are also, as might be expected
in a work of this sort, a (small) number of simplifications of proofs, etc.
No attempt, however, has been made to take
specific notice of such minor improvements.
At times I have
f'o.lLowed the papers of Melnikov and Vitushkin quite closely; in other instances, the original material has been reorganized considerably.
The reader who consults the original papers
w'ill easily identify the sections in question. These notes are based in part on a lecture given at the Brandeis-Brown-M.l.T. joint function algebra seminar at Brown University and on a series of lectures given at Professor Kenneth Hoffman's function algebra seminar at M.l.T.
I would
like to thank Professor T. W. Gamelin, who first interested me in giving the series of talks mentioned above; his help, encouragement, and enthusiasm at every step of the way have been invaluable.
In particular, he read and criticized a
pr
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