Oversampling on a class of symmetric regular de Branges spaces

SILVA, LUIS O.; TOLOZA, JULIO H.

Complex Variables and Elliptic Equations

Taylor & Francis

Año: 2023 p. 1 - 20

1747-6933

A de Branges space $cB$ 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 $K_cB(z,w)$ be the reproducing kernel in $cB$ and $S_cB$ be the operator of multiplication by the independent variable with maximal domain in $cB$.Loosely speaking, we say that $cB$ has the $ell_p$-oversampling property relative to a proper subspace $cA$ of it, with $pin(2,infty]$, if there exists $J_{cAcB}:CimesCoC$ such that $J(cdot,w)incB$ for all $winC$,[sum_{lambdainsigma(S_{cB}^{gamma})}left(rac{abs{J_{cAcB}(z,lambda)}}{K_cB(lambda,lambda)^{1/2}}ight)^{p/(p-1)}<infty;,ext{and};,F(z) = sum_{lambdainsigma(S_{cB}^{gamma})}        rac{J_{cAcB}(z,lambda)}{K_cB(lambda,lambda)}F(lambda),]for all {color{red}$FincA$} and almost every self-adjoint extension$S_{cB}^{gamma}$ of $S_cB$. 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.