Cardinal Interpolation with Gaussian Kernels

TL;DR

This paper investigates Gaussian kernel-based interpolation, providing $L_p$ Sobolev error estimates and showing the error's dependence on Fourier multiplier norms, with bounds independent of grid spacing for certain $p$ values.

Contribution

It establishes new $L_p$ Sobolev error bounds for Gaussian kernel interpolation and relates the error to Fourier multiplier norms, extending understanding of interpolation accuracy.

Findings

Error bounds are independent of grid spacing for $1<p< finite$

Error estimates involve Fourier multiplier norms related to the Hilbert transform

Logarithmic factors appear in bounds when $p=1$ or $ finite$

Abstract

In this paper, interpolation by scaled multi-integer translates of Gaussian kernels is studied. The main result establishes $L_p$ Sobolev error estimates and shows that the error is controlled by the $L_p$ multiplier norm of a Fourier multiplier closely related to the cardinal interpolant, and comparable to the Hilbert transform. Consequently, its multiplier norm is bounded independent of the grid spacing when $1<p<\infty$, and involves a logarithmic term when $p=1$ or $\infty$.