A Note on Regularized Shannon's Sampling Formulae

TL;DR

This work provides a rigorous $L^2$ error analysis for regularized Shannon's sampling formulae used to interpolate and differentiate bandlimited functions, grounding previously empirical DSC-based methods in theory. Under the assumptions that $f \in L^2(\mathbb{R}) \cap C^{s}(\mathbb{R})$ is bandlimited to $B$ and a Hermite-weighted regularization is applied, the authors derive a concrete error bound that decomposes into a regularization term and two exponentially decaying truncation terms, with the bound expressed in terms of $\Delta$, $\sigma$, $B$, $M_1$, and $M_2$. The proof hinges on a decomposition of the error into $E_1$, $E_2$, and $E_3$, Fourier-domain analysis for the regularization error, and Abel-type inequalities together with monotonicity lemmas to control the truncation tails. The results provide explicit guidance for choosing parameters to achieve a desired accuracy, supporting the robust use of regularized Shannon's formulae in interpolations, derivatives, and PDE solvers with improved accuracy and efficiency.

Abstract

Error estimation is given for a regularized Shannon's sampling formulae, which was found to be accurate and robust for numerically solving partial differential equations.