Related papers: A Hilbert space of Dirichlet series and systems of…

Inspired by the work of Hedenmalm, Lindqvist and Seip, we consider different properties of dilations systems of a fixed function $\varphi \in L^2(0,1)$. More precisely, we study when the system $\{\varphi(nx)\}_n$ is a Bessel sequence, a…

For a real number $\alpha$ the Hilbert spaces $\mathscr{D}_\alpha$ 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…

In this paper we formulate a concrete method for determining whether a system of dilated periodic functions forms a Riesz basis in $L^2(0,1)$. This method relies on a general framework developed by Hedenmalm, Lindqvist and Seip about 20…

Given a frequency $\lambda=(\lambda_n)$, we study when almost all vertical limits of a $\mathcal{H}_1$-Dirichlet series $\sum a_n e^{-\lambda_ns}$ are Riesz-summable almost everywhere on the imaginary axis. Equivalently, this means to…

The Dirichlet--Hardy space $\Ht$ 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 half-plane $\operatorname{Re} s>1/2$ is a necessary and sufficient…

We examine the threshold of the cyclicity for functions in Dirichlet-type spaces $\mathcal{D}_{\alpha}$, $\alpha\in(0,1]$. Given a fixed $\alpha^{*}\in(0,1]$, we construct a holomorphic function $f\in\mathcal{D}_{\alpha^{*}}$ which is…

The aim of these notes is to discuss the completeness of the dilated systems in a most general framework of an arbitrary sequence lattice $X$, including weighted $\ell^p$ spaces. In particular, general multiplicative and completely…

In this paper, certain classes of Hilbert spaces of Dirichlet series with weighted norms and their corresponding multiplier algebras will be explored. For a sequence $\{w_n\}_{n=n_0}^\infty $ of positive numbers, define \[\mathcal…

For a sequence $\mathbf w = \{w_j\}_{j = 2}^\infty$ of positive real numbers, consider the positive semi-definite kernel $\kappa_{\mathbf w}(s, u) = \sum_{j = 2}^\infty w_j j^{-s - \overline{u}}$ defined on some right-half plane $\mathbb…

We characterize the space of multipliers from the Hardy space of Dirichlet series $\mathcal H_p$ into $\mathcal H_q$ for every $1 \leq p,q \leq \infty$. For a fixed Dirichlet series, we also investigate some structural properties of its…

A weighted Hilbert space $F^2_{\varphi}$ of entire functions of $n$ variables is considered in the paper. The weight function $\varphi$ is a convex function on ${\mathbb C}^n$ depending on modules of variables and growing at infinity faster…

We introduce a large family of reproducing kernel Hilbert spaces $\mathcal{H} \subset \mbox{Hol}(\mathbb{D})$, which include the classical Dirichlet-type spaces $\mathcal{D}_\alpha$, by requiring normalized monomials to form a Riesz basis…

Given a frequency $\lambda = (\lambda_n)$ and $\ell \ge 0$, we introduce the scale of Banach spaces $H_{\infty,\ell}^{\lambda}[Re > 0]$ of holomorphic functions $f$ on the open right half-plane $[Re > 0]$, which satisfy $(A)$ the growth…

Given a periodic function $f$, we study the convergence almost everywhere and in norm of the series $\sum_{k} c_k f(kx)$. Let $f(x)= \sum_{m=1}^\infty a_m \sin {2\pi m x}$ where $\sum_{m=1}^\infty a_{m }^2d(m) <\infty$ and $d(m)=\sum_{d|m}…

The harmonic Dirichlet space $\cal{D} (\mathbb{T})$ is the Hilbert space of functions $f \in L^2(\mathbb{T})$ such that $$\|f\|_{\cal{D} (\mathbb{T})}^2 := \sum_{n\in\mathbb{Z}} (1+|n|)|\hat{f}(n)|^2 < \infty.$$ We give sufficient…

By a theorem of Gordon and Hedenmalm, $\varphi$ generates a bounded composition operator on the Hilbert space $\mathscr{H}^2$ of Dirichlet series $\sum_n b_n n^{-s}$ with square-summable coefficients $b_n$ if and only if $\varphi(s)=c_0…

We provide improved sufficient assumptions on sequences of Fucik eigenvalues of the one-dimensional Dirichlet Laplacian which guarantee that the corresponding Fucik eigenfunctions form a Riesz basis in $L^2(0,\pi)$. For that purpose, we…

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form $k(s,u) = \sum a_n n^{-s-\bar u}$, and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing…

Consider the Hill operator $L(v) = - d^2/dx^2 + v(x) $ on $[0,\pi]$ with Dirichlet, periodic or antiperiodic boundary conditions; then for large enough $n$ close to $n^2 $ there are one Dirichlet eigenvalue $\mu_n$ and two periodic (if $n$…

This paper concerns a long-standing problem raised by Beurling and Wintner on completeness of the dilation system $\{\varphi(kx):k=1,2,\cdots\}$ generated by the odd periodic extension on $\mathbb{R}$ of any $\varphi\in L^2[0,1]$. Up to now…