Analytic Functions Smooth up to the Boundary
This research monograph concerns the Nevanlinna factorization of analytic functions smooth, in a sense, up to the boundary. The peculiar properties of such a factorization are investigated for the most common classes of Lipschitz-like analytic functions.
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1312 Nikolai A. Shirokov
Analytic Functions Smooth up to the Boundary
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Nikolai A. Shirokov Fontanka 27, LOMI 191011 Leningrad, 0·11, USSR Consulting Editor
Sergei V. Khrushchev Fontanka 27, LOMI 191011 Leningrad, 0·11, USSR
Mathematics Subject Classification (1980): 30050, 46J20 ISBN 3-540·19255·7 Sprinqer-Verlaq Berlin Heidelberg New York ISBN 0-387-19255-7 Sprinqer-Verlaq New York Berlin Heidelberg
Library of Congress Cataloging-in-Publication Data. Shirokov, Nikolai A., 1948- Analytic functions smooth up to the boundary. (Lecture notes in mathematics; 1312) "Subseries: USSR." Bibliography: p. Includes index. 1. Analytic functions. 2. Multipliers (Mathematical analysis) I. Title. II. Series: Lecture notes in mathematics (Sprinqer-Verlaq): 1312. 0A3.L28 no. 1312 [0A331] 510 s [515J 88-12336 ISBN 0-387·19255-7 (U.S.) This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1988 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
CONTENTS
Introduction .•••.....•••..•....•.....•...............•...•..••.. Notations . . . . . . . • . . • . • . . . . . • . • . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . .
21
Chapter I. The
.
25
••••••••••••••••••••
25
§ 1. The
(F) -property
( F) -property
for
A't"vw ( 0,
is the following: a function
iff the singular measure
f
be-
is nonnegative. Thus
the Hardy classes can be characterized as follows:
5
3. Let us now dwell on some details connected with the formula (2) which are especially important for our paper. I. We say that an inner function
The above references yield that if 1 divides then {r !O
I
if an inner function
t
if
HP
the function
t
I
and
divides an inner function
If
14
divides
JI-/-
t
4r'
then
f
is obtained from
As a matter of fact the function
J r °t
since
ID
has no zeros in
HP
Similarly
Roughly speaking
by "removing the zeros" of 1 does not vanish in ,
{r
and removing of
sense) an isolation of the "boundary zeros" of {
and
t.
means (in a The outer factor
behaves in ApproXimation Theory and Theory of Invariant Subspaces
in many respects as invertible. Thus
m
and
HP
are invariant with
respect to the "isolation of zeros". IT. Nevanlinna's theorem contains the complete information about moduli of functions from For example, let
h,
N
or
HP
on . Then the
be a nonnegative function on
following statements are equivalent:
(ol.)
t01
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