Sampling and Galerkin reconstruction in reproducing kernel spaces
This work provides a theoretical framework for quasi-optimal signal reconstruction from samples in reproducing kernel spaces, which is relevant for signal processing and approximation theory.
The paper introduces a Galerkin reconstruction method for sampling in reproducing kernel subspaces of L^p, achieving quasi-optimal approximation with an iterative algorithm, and demonstrates its effectiveness on signals with finite rate of innovation.
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.