Sampling and Galerkin reconstruction in reproducing kernel spaces

TL;DR

This paper introduces a Galerkin reconstruction method for sampling in reproducing kernel subspaces of L^p, providing a quasi-optimal approximation and an iterative solution approach, with analysis and simulations for signals with finite rate of innovation.

Contribution

It proposes a novel Galerkin reconstruction framework in Banach spaces for sampling in reproducing kernel spaces, including an iterative solution method.

Findings

The Galerkin method achieves quasi-optimal approximation.

The iterative algorithm effectively solves the Galerkin equations.

Numerical simulations demonstrate successful reconstruction of signals with finite rate of innovation.

Abstract

In this paper, we consider sampling in a reproducing kernel subspace of $L^p$. We introduce a pre-reconstruction operator associated with a sampling scheme and propose a Galerkin reconstruction in general Banach space setting. We show that the proposed Galerkin method provides a quasi-optimal approximation, and the corresponding Galerkin equations could be solved by an iterative approximation-projection algorithm. We also present detailed analysis and numerical simulations of the Galerkin method for reconstructing signals with finite rate of innovation.