# A Hilbert space of Dirichlet series and systems of dilated functions in $L^2(0,1)$

@article{Hedenmalm1995AHS,
title={A Hilbert space of Dirichlet series and systems of dilated functions in \$L^2(0,1)\$},
author={H̊akan Hedenmalm and Peter Lindqvist and Kristian Seip},
journal={Duke Mathematical Journal},
year={1995},
volume={86},
pages={1-37}
}
• Published 1 December 1995
• Mathematics
• Duke Mathematical Journal
For a function $\varphi$ in $L^2(0,1)$, extended to the whole real line as an odd periodic function of period 2, we ask when the collection of dilates $\varphi(nx)$, $n=1,2,3,\ldots$, constitutes a Riesz basis or a complete sequence in $L^2(0,1)$. The problem translates into a question concerning multipliers and cyclic vectors in the Hilbert space $\cal H$ of Dirichlet series $f(s)=\sum_n a_nn^{-s}$, where the coefficients $a_n$ are square summable. It proves useful to model $\cal H$ as the $H… 170 Citations Zeros of functions in Hilbert spaces of Dirichlet series The Dirichlet–Hardy space $${\fancyscript{H}}^2$$ consists of those Dirichlet series $$\sum _n a_n n^{-s}$$ for which $$\sum _n |a_n|^2<\infty$$. It is shown that the Blaschke condition in the Hardy–Orlicz Spaces of Dirichlet Series: An Interpolation Problem on Abscissae of Convergence • Mathematics International Mathematics Research Notices • 2019 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 Riesz means in Hardy spaces on Dirichlet groups • Mathematics Mathematische Annalen • 2020 Given a frequency $$\lambda =(\lambda _n)$$ λ = ( λ n ) , we study when almost all vertical limits of a $$\mathcal {H}_1$$ H 1 -Dirichlet series $$\sum a_n e^{-\lambda _ns}$$ ∑ a n e - λ n s are Zeros of Functions in Bergman-Type Hilbert Spaces of Dirichlet Series 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$\sum_{n=1}^\infty |a_n|^2/[d(n)]^\alpha 0$. Generalizations Hardy spaces of vector-valued Dirichlet series • Mathematics • 2016 Given a Banach space$X$and$1 \leq p \leq \infty$, it is well known that the two Hardy spaces$H_p(\mathbb{T},X)$($\mathbb{T}$the torus) and$H_p(\mathbb{D},X)$($\mathbb{D}$the disk) have to be Composition Operators on Bohr-Bergman Spaces of Dirichlet Series • Mathematics • 2016 For$\alpha \in \mathbb{R}$, let$\mathscr{D}_\alpha$denote the scale of Hilbert spaces consisting of Dirichlet series$f(s) = \sum_{n=1}^\infty a_n n^{-s}$that satisfy$\sum_{n=1}^\infty
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## References

SHOWING 1-10 OF 21 REFERENCES
Foundations of the theory of Dirichlet series
1. A modern reader, familiar with the methods of functional analysis, is struck with the conviction tha t the classical theory of Dirichlet series [3, 5, 6] must have content expressible in more
An introduction to nonharmonic Fourier series
Bases in Banach Spaces - Schauder Bases Schauder's Basis for C[a,b] Orthonormal Bases in Hilbert Space The Reproducing Kernel Complete Sequences The Coefficient Functionals Duality Riesz Bases The
An Introduction to the Theory of Numbers
• Philosophy
• 1938
This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford,
On the convergence of multiple Fourier series
Inequality (1) follows from the special case in which P is a triangle with a vertex at the origin; for any polygon breaks up into triangles, and the characteristic function of any triangle is a
What is ergodic theory
Ergodic theory involves the study of transformations on measure spaces. Interchanging the words “measurable function” and “probability density function” translates many results from real analysis to
Über die gleichmäßige Konvergenz Dirichletscher Reihen.
Einleitung. Es sei (1.) 0< 1< 2<·. .< <... (lim . = ) n=oo eine Folge reeller Zahlen und (2.) iane-» n=l eine zugehörige Dirichletsche Reihe, die ein Konvergenzgebiet besitzt. Dann existieren
Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe
[Aus dem dreizehnten Bande der Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen.]