On the boundary behaviour of the Hardy spaces of Dirichlet series and a frame bound estimate

2012 ◽  
Vol 2012 (663) ◽  
Author(s):  
Jan-Fredrik Olsen ◽  
Eero Saksman
Keyword(s):  
Dirichlet Series ◽  
Hardy Spaces ◽  
Boundary Behaviour ◽  
Frame Bound

Hardy–Orlicz Spaces of Dirichlet Series: An Interpolation Problem on Abscissae of Convergence

2019 ◽  
Author(s):  
Maxime Bailleul ◽  
Pascal Lefèvre ◽  
Luis Rodríguez-Piazza
Keyword(s):  
Dirichlet Series ◽  
Hardy Spaces ◽  
Upper Bound ◽  
Interpolation Problem ◽  
Orlicz Spaces ◽  
Elementary Method ◽  
Abscissa Of Convergence ◽  
Upper Bound Estimate

Abstract The study of Hardy spaces of Dirichlet series denoted by $\mathscr{H}^p$ ($p\geq 1$) was initiated in [7] when $p=2$ and $p=\infty $, and in [2] for the general case. In this paper we introduce the Orlicz version of spaces of Dirichlet series $\mathscr{H}^\psi $. We focus on the case $\psi =\psi _q(t)=\exp (t^q)-1,$ and we compute the abscissa of convergence for these spaces. It turns out that its value is $\min \{1/q\,,1/2\}$ filling the gap between the case $\mathscr{H}^\infty $, where the abscissa is equal to $0$, and the case $\mathscr{H}^p$ for $p$ finite, where the abscissa is equal to $1/2$. The upper-bound estimate relies on an elementary method that applies to many spaces of Dirichlet series. This answers a question raised by Hedenmalm in [6].

scholarly journals Zeros of Functions in Bergman-Type Hilbert Spaces of Dirichlet Series

2016 ◽  
Vol 119 (2) ◽  
pp. 237
Author(s):  
Ole Fredrik Brevig
Keyword(s):  
Hilbert Space ◽  
Real Number ◽  
Dirichlet Series ◽  
Hilbert Spaces ◽  
Hardy Spaces ◽  
Weighted Spaces ◽  
Number Of Divisors ◽  
Zeros Of Functions

For a real number $\alpha$ the Hilbert space $\mathscr{D}_\alpha$ consists of those Dirichlet series $\sum_{n=1}^\infty a_n/n^s$ for which $\sum_{n=1}^\infty |a_n|^2/[d(n)]^\alpha < \infty$, where $d(n)$ denotes the number of divisors of $n$. We extend a theorem of Seip on the bounded zero sequences of functions in $\mathscr{D}_\alpha$ to the case $\alpha>0$. Generalizations to other weighted spaces of Dirichlet series are also discussed, as are partial results on the zeros of functions in the Hardy spaces of Dirichlet series $\mathscr{H}^p$, for $1\leq p <2$.

scholarly journals Fourier Multipliers for Hardy Spaces of Dirichlet Series

2013 ◽  
Vol 2014 (16) ◽  
pp. 4368-4378 ◽  
Author(s):  
Alexandru Aleman ◽  
Jan-Fredrik Olsen ◽  
Eero Saksman
Keyword(s):  
Dirichlet Series ◽  
Hardy Spaces ◽  
Fourier Multipliers

Interpolation of Hardy spaces of Dirichlet series

2019 ◽  
Vol 277 (3) ◽  
pp. 786-805
Author(s):  
Frédéric Bayart ◽  
Mieczysław Mastyło
Keyword(s):  
Dirichlet Series ◽  
Hardy Spaces

scholarly journals Hardy spaces of general Dirichlet series — a survey

2019 ◽  
Vol 119 ◽  
pp. 123-149 ◽  
Author(s):  
Andreas Defant ◽  
Ingo Schoolmann
Keyword(s):  
Dirichlet Series ◽  
Hardy Spaces ◽  
General Dirichlet Series

Hardy spaces of Dirichlet Series

2013 ◽  
pp. 139-184
Author(s):  
Hervé Queffélec ◽  
Martine Queffélec
Keyword(s):  
Dirichlet Series ◽  
Hardy Spaces
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