Exponential Approximation of Band-Limited Functions from Nonuniform Sampling by Regularization Methods

Reconstructing a band-limited function from its finite sample data is a fundamental task in signal analysis. A Gaussian regularized Shannon sampling series has been proven to be able to achieve exponential convergence for uniform sampling. In this paper, we prove that such an exponential convergence can also be achieved for nonuniform sampling by regularization methods. Specifically, it is shown that one can recover a band-limited function by Gaussian or hyper-Gaussian regularized nonuniform sampling series with an explicit exponential convergence rate. The analysis is based on the residue theorem in complex analysis to express the truncation error by a contour integral, and the Laplace method to estimate integrals. Several concrete examples of nonuniform sampling with exponential convergence will be presented.