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The Capacity of Sets of Divergence of Certain Taylor Series on the Unit Circle
A simple and direct proof is given of a generalization of a classical result on the convergence of ∑ k = 0 ∞ a k e ikx outside sets of x of an appropriate capacity zero, where f ( z ) = ∑ k = 0 ∞ a k z k is analytic in the unit disc U and ∑ k = 0 ∞ k α | a k | 2 < ∞ with α ∈ ( 0 , 1 ] . We also d...
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| Published in: | Computational methods and function theory 2019-06, Vol.19 (2), p.227-236 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | A simple and direct proof is given of a generalization of a classical result on the convergence of
∑
k
=
0
∞
a
k
e
ikx
outside sets of
x
of an appropriate capacity zero, where
f
(
z
)
=
∑
k
=
0
∞
a
k
z
k
is analytic in the unit disc
U
and
∑
k
=
0
∞
k
α
|
a
k
|
2
<
∞
with
α
∈
(
0
,
1
]
.
We also discuss some convergence consequences of our results for weighted Besov spaces, for the classes of analytic functions in
U
for which
∑
k
=
1
∞
k
γ
|
a
k
|
p
<
∞
,
and for trigonometric series of the form
∑
k
=
1
∞
(
α
k
cos
k
x
+
β
k
sin
k
x
)
with
∑
k
=
1
∞
k
γ
(
|
α
k
|
p
+
|
β
k
|
p
)
<
∞
, where
γ
>
0
,
p
>
1
. |
|---|---|
| ISSN: | 1617-9447 2195-3724 |
| DOI: | 10.1007/s40315-019-00266-z |