Oversampling on a class of symmetric regular de Branges spaces

Abstract

A de Branges space $\mathcal{B}$ is regular if the constants belong to its space of associated functions and is symmetric if it is isometrically invariant under the map $F(z)\mapsto F(-z)$. Let $K_{\mathcal{B}}(z,w)$ be the reproducing kernel in $\mathcal{B}$ and $S_{\mathcal{B}}$ be the operator of multiplication by the independent variable with maximal domain in $\mathcal{B}$. Loosely speaking, we say that $\mathcal{B}$ has the ${\ell }_p$-oversampling property relative to a proper subspace $\mathcal{A}$ of it, with $p\in (2,\infty ]$, if there exists $J_{\mathcal{A}\mathcal{B}}:\mathbb{C}\times \mathbb{C}\to \mathbb{C}$ such that $J(\cdot ,w)\in \mathcal{B}$ for all $w\in \mathbb{C}$, \begin{equation*} \sum_{λ\inσ(S_{\mathcal B}^γ)} \left(\frac{\lvert J_{\mathcal{A}\mathcal{B}}(z,λ)\rvert}{K_\mathcal{B}(λ,λ)^{1/2}}\right)^{p/(p-1)} <\infty, \quad\text{and}\quad F(z) = \sum_{λ\inσ(S_{\mathcal B}^γ)} \frac{J_{\mathcal{A}\mathcal{B}}(z,λ)}{K_\mathcal{B}(λ,λ)}F(λ), \end{equation*} for all $F\in \mathcal{A}$ and almost every self-adjoint extension $S_{\mathcal{B}}^{\gamma }$ of $S_{\mathcal{B}}$. This definition is motivated by the well-known oversampling property of Paley-Wiener spaces. In this paper we provide sufficient conditions for a symmetric, regular de Branges space to have the ${\ell }_p$-oversampling property relative to a chain of de Branges subspaces of it.

18 pages, some minor corrections