Date of Award

2026

Document Type

Honors Thesis (Open Access)

Department

Colby College. Mathematics and Statistics Dept.

Advisor(s)

Evan Randles

Second Advisor

Leo Livshits

Abstract

In this thesis we provide Gaussian Estimates and Local Limit Theorems describing the asymptotic behavior of convolution powers of a class of complex-valued functions on $\mathbb{Z}^d$. Convolution powers arise naturally in the study of partial differential equations, as well as in random walks in probability theory. In particular, they are connected to the stability theory of difference schemes used to approximate solutions to partial differential equations. We take inspiration from the work of Vidar Thomée on stability theory to restrict our attention to convolution powers of functions whose Fourier Transforms satisfy certain local expansions. We then combine the Cauchy Integral Theorem from complex analysis and an integral identity for convolution powers to establish our main results, building on recent work from J.-F. Coulombel and G. Faye. We also show how our theorems apply to stability theory. The main results presented here appear in a forthcoming article by E. Randles, J. M. Valdovinos, and the author.

Keywords

Convolution powers, Gaussian estimates, local limit theorems, stability of numerical difference schemes

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