On numerical realizations of Shannon's sampling theorem

TL;DR

This paper analyzes the numerical implementation of Shannon's sampling theorem, highlighting convergence issues, and proposes oversampling and regularization techniques with specific window functions to improve approximation accuracy.

Contribution

It compares frequency and time domain regularization methods, demonstrating that time domain oversampling with sinh-type or Kaiser-Bessel windows yields superior results.

Findings

Classical Shannon sampling sums have poor convergence.

Oversampling and regularization improve approximation accuracy.

Time domain window functions outperform frequency domain methods.

Abstract

In this paper, we discuss some numerical realizations of Shannon's sampling theorem. First we show the poor convergence of classical Shannon sampling sums by presenting sharp upper and lower bounds of the norm of the Shannon sampling operator. In addition, it is known that in the presence of noise in the samples of a bandlimited function, the convergence of Shannon sampling series may even break down completely. To overcome these drawbacks, one can use oversampling and regularization with a convenient window function. Such a window function can be chosen either in frequency domain or in time domain. We especially put emphasis on the comparison of these two approaches in terms of error decay rates. It turns out that the best numerical results are obtained by oversampling and regularization in time domain using a sinh-type window function or a continuous Kaiser-Bessel window function, which results in an interpolating approximation with localized sampling. Several numerical experiments illustrate the theoretical results.