An Optimal Convergence Rate for the Gaussian Regularized Shannon Sampling Series

TL;DR

This paper proves that the Gaussian regularized Shannon sampling series achieves the optimal convergence rate for reconstructing bandlimited functions from finite samples, improving upon traditional methods.

Contribution

It provides the first theoretical proof that the Gaussian regularization method attains the best possible convergence rate for certain bandwidths.

Findings

The convergence rate of the Gaussian regularized Shannon series is optimal for bandwidths less than π/2.

Numerical experiments confirm the theoretical convergence rate.

The method significantly outperforms classical Shannon sampling in convergence speed.

Abstract

We consider the reconstruction of a bandlimited function from its finite localized sample data. Truncating the classical Shannon sampling series results in an unsatisfactory convergence rate due to the slow decay of the sinc function. To overcome this drawback, a simple and highly effective method, called the Gaussian regularization of the Shannon series, was proposed in the engineering and has received remarkable attention. It works by multiplying the sinc function in the Shannon series with a regularized Gaussian function. Recently, it was proved that the upper error bound of this method can achieve a convergence rate of the order $O(\frac{1}{\sqrt{n}}\exp(-\frac{\pi-\delta}{2}n))$, where $0<\delta<\pi$ is the bandwidth and $n$ the number of sample data. The convergence rate is by far the best convergence rate among all regularized methods for the Shannon sampling series. The main objective of this article is to present the theoretical justification and numerical verification that the convergence rate is optimal when $0<\delta<\pi/2$ by estimating the lower error bound of the truncated Gaussian regularized Shannon sampling series.