A converse sampling theorem in reproducing kernel Banach spaces

Abstract

We present a converse Kramer type sampling theorem over semi-inner product reproducing kernel Banach spaces. Assuming that a sampling expansion holds for every f belonging to a semi-inner product reproducing kernel Banach space B for a fixed sequence of interpolating functions {aj-1Sj(t)}j and a subset of sampling points {tj}j, it results that such sequence must be a Xd∗-Riesz basis and a sampling basis for the space. Moreover, there exists an equivalent (in norm) reproducing kernel Banach space with a reproducing kernel Gsamp such that {a¯j-1Gsamp(tj,.)}j and {aj-1Sj(.)}j are biorthogonal. These results are a generalization of some known results over reproducing kernel Hilbert spaces.