Convolution powers of complex functions on $\mathbb{Z}^d$

TL;DR

This paper investigates the behavior of convolution powers of complex-valued functions on multi-dimensional integer lattices, extending classical probability and stability results to a more general complex setting with applications in PDEs.

Contribution

It extends existing theories of convolution powers from probability distributions to complex functions on $ abla^d$, addressing fundamental questions in sup-norm estimates, local limit theorems, and stability.

Findings

Extended local limit theorems for complex functions.

Derived sup-norm estimates for convolution powers.

Analyzed stability conditions for complex-valued convolutions.

Abstract

The study of convolution powers of a finitely supported probability distribution $\phi$ on the $d$-dimensional square lattice is central to random walk theory. For instance, the $n$th convolution power $\phi^{(n)}$ is the distribution of the $n$th step of the associated random walk and is described by the classical local limit theorem. Following previous work of P. Diaconis and the authors, we explore the more general setting in which $\phi$ takes on complex values. This problem, originally motivated by the problem of Erastus L. De Forest in data smoothing, has found applications to the theory of stability of numerical difference schemes in partial differential equations. For a complex valued function $\phi$ on $\mathbb{Z}^d$, we ask and address four basic and fundamental questions about the convolution powers $\phi^{(n)}$ which concern sup-norm estimates, generalized local limit theorems, pointwise estimates, and stability. This work extends one-dimensional results of I. J. Schoenberg, T. N. E. Greville, P. Diaconis and the second author and, in the context of stability theory, results by V. Thom\'ee and M. V. Fedoryuk.