Sampling theory in a general reproducing kernel Hilbert space

Abstract

Sampling procedure consists of the reduction of an analog signal into a digital signal and the reconstruction of the original signal from its discrete values. Starting from the classical WSK theorem, various extensions in sampling theory have been developed and widely applied in signal processing and information theory. This dissertation handles the sampling expansions in a general reproducing kernel Hilbert space. We begin by introducing engineering approach of WSK theorem in a Paley-wiener space. Then we describe the general sampling theorem in a reproducing kernel Hilbert space setting which is a subspace of $L^2(\mathbb{R})$ closed in a particular sobolev space and develop the theorems with more general conditions. Secondly, we deal with a construction of a reproducing kernel Hilbert space which admits a stable sampling set and characterize its properties. Finally we draw sampling expansions in the constructed space.