Sharp exponential bounds for the Gaussian regularized Whittaker-Kotelnikov-Shannon sampling series

Fast reconstruction of a bandlimited function from its finite oversampling data has been a fundamental problem in sampling theory. As the number of sample data increases to infinity, exponentially-decaying reconstruction errors can be achieved by many methods in the literature. In fact, it is generally conjectured that when the optimal method is used, the dominant term in the error of reconstructing a function bandlimited to [-delta, delta] (delta < pi) from its data sampled at the integer points on [-n, n] is exp(-lambda(pi - delta)n). By far, the best estimate for the constant lambda among regularization methods is 1/2 and is achieved by the highly efficient Gaussian regularized Whittaker-Kotelnikov-Shannon sampling series. We prove in this paper that the exponential constant 1/2 is optimal for this method. Moreover, the optimal variance of the Gaussian regularizer is provided. (C) 2019 Elsevier Inc. All rights reserved.