Optimal parameter choice for regularized Shannon sampling formulas

The fast reconstruction of a bandlimited function from its sample data is an essential problem in signal processing. In this paper, we consider the widely used Gaussian regularized Shannon sampling formula in comparison to regularized Shannon sampling formulas employing alternative window functions, including the modified Gaussian function, the sinh-type window function, and the continuous Kaiser-Bessel window function. It is shown that the approximation errors of these regularized Shannon sampling formulas possess an exponential decay with respect to the truncation parameter. The main focus of this paper is to identify the optimal variance of the (modified) Gaussian function as well as the optimal shape parameters of the sinh-type window function and the continuous Kaiser-Bessel window function, with the aim of achieving the fastest exponential decay of the approximation error. In doing so, we demonstrate that the decay rate of the sinh-type regularized Shannon sampling formula is considerably superior to that of the Gaussian regularized Shannon sampling formula. Additionally, numerical experiments illustrate the theoretical results.