Oversampling on a class of symmetric regular de Branges spaces

Abstract

A de Branges space 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)↦F(−z) . Let KB(z,w) be the reproducing kernel in B and SB be the operator of multiplication by the independent variable with maximal domain in B . Loosely speaking, we say that B has the ℓp -oversampling property relative to a proper subspace A of it, with p∈(2,∞] , if there exists JAB:C×C→C such that J(⋅,w)∈B for all w∈C , ∑λ∈σ(SγB)(|JAB(z,λ)|KB(λ,λ)1/2)p/(p−1)<∞andF(z)=∑λ∈σ(SγB)JAB(z,λ)KB(λ,λ)F(λ), for all F∈A and almost every self-adjoint extension SγB of SB . 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 ℓp -oversampling property relative to a chain of de Branges subspaces of it.