Oversampling on a class of symmetric regular de Branges spaces

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) bar right arrow F(-z). Let K-B(z, w) be the reproducing kernel in B and S-B be the operator of multiplication by the independent variable with maximal domain in B. Loosely speaking, we say that B has the l(p)-oversampling property relative to a proper subspace A of it, with p is an element of (2, infinity], if there exists J(AB) : C x C -> C such that J(center dot, w) is an element of B for all w is an element of C, Sigma(lambda is an element of sigma (SBP)) (vertical bar J(AB)(Z, lambda)vertical bar/K-B(lambda, lambda)(1/2))(p/(p-1)) < infinity and F(Z) = Sigma(lambda is an element of sigma (SB gamma)) (vertical bar J(AB)(Z, lambda)vertical bar/K-B(lambda, lambda)F(lambda), for all F is an element of A and almost every self-adjoint extension S-B(gamma) of S-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 l(p)-oversampling property relative to a chain of de Branges subspaces of it.