A converse sampling theorem in reproducing kernel Banach spaces

Abstract

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 xed sequence of interpolating functions {a −1 j Sj (t)}j and a subset of sampling points {tj}j , it results that such sequence must be a X∗ d -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 −1 j Gsamp(tj , .)}j and {a −1 j Sj (.)}j are biorthogonal. These results are a generalization of some known results over reproducing kernel Hilbert spaces.