Radial limit sets on the torus
Author:
Laurence D. Hoffmann
Journal:
Trans. Amer. Math. Soc. 171 (1972), 283-290
MSC:
Primary 43A70; Secondary 32E25, 46J15
DOI:
https://doi.org/10.1090/S0002-9947-1972-0330934-5
MathSciNet review:
0330934
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Abstract | References | Similar Articles | Additional Information
Abstract: Let denote the unit polydisc and
the unit torus in the space of
complex variables. A subset
of
is called an (RL)-set (radial limit set) if to each positive continuous function
on
, there corresponds a function
in
such that the radial limit
of the absolute value of
equals
, a.e. on
and everywhere on
. If
, the question of characterizing (RL)-sets is open, but two positive results are obtained. In particular, it is shown that
contains an (RL)-set which is homeomorphic to a cartesian product
, where
is a Cantor set. Also, certain countable unions of ``parallel'' copies of
are shown to be (RL)-sets in
. In one variable, every subset of
is an (RL)-set; in fact, there is always a zero-free function
in
with the required properties. It is shown, however, that there exist a circle
and a positive continuous function
on
to which correspond no zero-free
in
with
a.e. on
and everywhere on
.
- [1] Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR 0133008
- [2] Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210528
- [3] Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
- [4] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1972-0330934-5
Keywords:
Polydisc,
torus,
Hardy space ,
radial limits,
radial limit sets,
outer functions,
inner functions
Article copyright:
© Copyright 1972
American Mathematical Society