Generalized Gaussian Estimates and Local Limit Theorems for Discrete Convolution Powers of Complex Functions: The $d$-dimensional case

TL;DR

This paper develops generalized Gaussian bounds and local limit theorems for the convolution powers of complex functions on multi-dimensional integer lattices, with applications to numerical schemes for PDEs.

Contribution

It extends existing results to the d-dimensional case, providing sharp estimates using Legendre-Fenchel transforms and connecting to heat kernel analysis.

Findings

Established sharp Gaussian bounds for convolution powers in multiple dimensions.

Derived local limit theorems with Gaussian-type error estimates.

Applied results to analyze stability of numerical difference schemes for PDEs.

Abstract

We establish generalized Gaussian bounds and local limit theorems with Gaussian-type error for the convolution powers of certain complex-valued functions on $\mathbb{Z}^d$. These global space-times estimates/error, which are sharp in certain cases, are written in terms of the Legendre-Fenchel transforms of positive-homogeneous polynomials and are mirrored by estimates satisfied by the heat kernels associated to a related class of partial differential operators. The results obtained here enjoy applications to the analysis and stability of numerical difference schemes to partial differential equations. This work extends several recent results, pertaining to one and several dimensions, of P. Diaconis, L. Saloff-Coste, J.-F. Coulombel, G. Faye, L. Coeuret, and the second author.