Some remarks on regularized Shannon sampling formulas

TL;DR

This paper compares various regularized Shannon sampling formulas, demonstrating that sinh-type and Kaiser-Bessel windows achieve exponential decay in approximation errors, with rigorous proofs and numerical validation.

Contribution

It provides rigorous proofs for the exponential decay rates of different regularized Shannon sampling formulas, clarifying previous assumptions and comparing their effectiveness.

Findings

Sinh-type and Kaiser-Bessel formulas have exponential decay similar to each other.

Both outperform Gaussian regularization in decay rate.

Numerical experiments confirm theoretical decay rates.

Abstract

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, such as 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 work is to address minor gaps in preceding papers and rigorously prove assumptions that were previously based solely on numerical tests. In doing so, we demonstrate that the sinh-type regularized Shannon sampling formula has the same exponential decay as the continuous Kaiser-Bessel regularized Shannon sampling formula, but both have twice the exponential decay of the Gaussian regularized Shannon sampling formula. Additionally, numerical experiments illustrate the theoretical results.