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Mean Rational Approximation for Some Compact Planar Subsets
Authors
John B. Conway
Liming Yang
Publication date
10 December 2022
Publisher
View
on
arXiv
Abstract
In 1991, J. Thomson obtained celebrated structural results for
P
t
(
μ
)
.
P^t(\mu).
P
t
(
μ
)
.
Later, J. Brennan (2008) generalized Thomson's theorem to
R
t
(
K
,
μ
)
R^t(K,\mu)
R
t
(
K
,
μ
)
when the diameters of the components of
C
∖
K
\mathbb C\setminus K
C
∖
K
are bounded below. The results indicate that if
R
t
(
K
,
μ
)
R^t(K,\mu)
R
t
(
K
,
μ
)
is pure, then
R
t
(
K
,
μ
)
∩
L
∞
(
μ
)
R^t(K,\mu) \cap L^\infty (\mu)
R
t
(
K
,
μ
)
∩
L
∞
(
μ
)
is the "same as" the algebra of bounded analytic functions on
\mbox{abpe}(R^t(K, \mu)),
the set of analytic bounded point evaluations. We show that if the diameters of the components of
C
∖
K
\mathbb C\setminus K
C
∖
K
are allowed to tend to zero, then even though
\text{int}(K) = \mbox{abpe}(R^t(K, \mu))
and
K
=
int
(
K
)
‾
,
K =\overline {\text{int}(K)},
K
=
int
(
K
)
​
,
the algebra
R
t
(
K
,
μ
)
∩
L
∞
(
μ
)
R^t(K,\mu) \cap L^\infty (\mu)
R
t
(
K
,
μ
)
∩
L
∞
(
μ
)
may "be equal to" a proper sub-algebra of bounded analytic functions on
int
(
K
)
,
\text{int}(K),
int
(
K
)
,
where functions in the sub-algebra are "continuous" on certain portions of the inner boundary of
K
.
K.
K
.
Comment: arXiv admin note: text overlap with arXiv:1904.0644
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oai:arXiv.org:2212.05392
Last time updated on 08/01/2023