  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.
