Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables
Let H ∞ be the set of all ordinary Dirichlet series D = ∑ n a n n - s representing bounded holomorphic functions on the right half plane. A completely multiplicative sequence ( b n ) of complex numbers is said to be an ℓ 1 -multiplier for H ∞ whenever ∑ n | a n b n | < ∞ for every D ∈ H ∞ . We st...
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| Veröffentlicht in: | Mathematische annalen Jg. 368; H. 1-2; S. 837 - 876 |
|---|---|
| Hauptverfasser: | , , , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.06.2017
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 0025-5831, 1432-1807 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | Let
H
∞
be the set of all ordinary Dirichlet series
D
=
∑
n
a
n
n
-
s
representing bounded holomorphic functions on the right half plane. A completely multiplicative sequence
(
b
n
)
of complex numbers is said to be an
ℓ
1
-multiplier for
H
∞
whenever
∑
n
|
a
n
b
n
|
<
∞
for every
D
∈
H
∞
. We study the problem of describing such sequences
(
b
n
)
in terms of the asymptotic decay of the subsequence
(
b
p
j
)
, where
p
j
denotes the
j
th prime number. Given a completely multiplicative sequence
b
=
(
b
n
)
we prove (among other results):
b
is an
ℓ
1
-multiplier for
H
∞
provided
|
b
p
j
|
<
1
for all
j
and
lim
¯
n
1
log
n
∑
j
=
1
n
b
p
j
∗
2
<
1
, and conversely, if
b
is an
ℓ
1
-multiplier for
H
∞
, then
|
b
p
j
|
<
1
for all
j
and
lim
¯
n
1
log
n
∑
j
=
1
n
b
p
j
∗
2
≤
1
(here
b
∗
stands for the decreasing rearrangement of
b
). Following an ingenious idea of Harald Bohr it turns out that this problem is intimately related with the question of characterizing those sequences
z
in the infinite dimensional polydisk
D
∞
(the open unit ball of
ℓ
∞
) for which every bounded and holomorphic function
f
on
D
∞
has an absolutely convergent monomial series expansion
∑
α
∂
α
f
(
0
)
α
!
z
α
. Moreover, we study analogous problems in Hardy spaces of Dirichlet series and Hardy spaces of functions on the infinite dimensional polytorus
T
∞
. |
|---|---|
| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0025-5831 1432-1807 |
| DOI: | 10.1007/s00208-016-1511-1 |