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

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. Thomée and M.V. Fedoryuk.