Hyper-Gaussian Regularized Whittaker–Kotel’nikov–Shannon Sampling Series

The reconstruction of a bandlimited function from its finite sample data is fundamental in signal analysis. It is well known that oversampling of a bandlimited function leads to exponential convergence in its reconstruction. A simple and efficient Gaussian regularized Shannon sampling formula has been proposed in G. W. Wei [Quasi wavelets and quasi interpolating wavelets, Chem. Phys. Lett. 296 (1998) 215–222] with such an exponential convergence ability. We show that all hyper-Gaussian regularized formulas share this desired property. The analysis is built on estimates on the Fourier transform of the hyper-Gaussian functions. We also establish error bounds for the reconstruction of derivatives of a univariate bandlimited function, and for multivariate bandlimited functions.