Approximation on disks
Author:
Kenneth John Preskenis
Journal:
Trans. Amer. Math. Soc. 171 (1972), 445-467
MSC:
Primary 41A20; Secondary 30A82
DOI:
https://doi.org/10.1090/S0002-9947-1972-0312123-3
MathSciNet review:
0312123
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Abstract | References | Similar Articles | Additional Information
Abstract: Let be a closed disk in the complex plane,
a complex valued continuous function on
and
the uniform closure on
of rational functions in
and
which are finite. Among other results we obtain the following. Theorem. If
is of class
in a neighborhood of
and
everywhere (i.e.,
is an orientation reversing immersion of
in the plane), then
. Theorem. Let
be a polynomial in
and
. If for each a in
with
at he zeros of
in
where
, then
. Corollary. Let
be a polynomial in
and
and let
. Then there exists an
such that, for
. The proofs of the theorems use measures and the conditions involved in the theorems are independent of each other. Concerning the corollary, results of E. Bishop and G. Stolzenberg show that
and
, then there exists no
such that
where
.
Let be a map on
= unit polydisk in
with values in
= uniform closure on
of polynomials in
. Theorem. If
is of class
in a neighborhood of
is invertible and if for each
in
, there exist complex constants
, such that
has positive real part for all
, then
is a polynomially convex set. Corollary. If
where
and the coefficients satisfy
and
, then
. Corollary. If
where
is of class
and satisfies the Lipschitz condition
with
, then
. This last corollary is a result of Hörmander and Wermer. The proof of the theorem uses methods from several complex variables.
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1972-0312123-3
Article copyright:
© Copyright 1972
American Mathematical Society