Convolution powers of complex functions on

  • Evan Randles

    Colby College, Waterville, USA
  • Laurent Saloff-Coste

    Cornell University, Ithaca, USA

Abstract

The study of convolution powers of a finitely supported probability distribution on the -dimensional square lattice is central to random walk theory. For instance, the th convolution power is the distribution of the 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 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 on , we ask and address four basic and fundamental questions about the convolution powers 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.

Cite this article

Evan Randles, Laurent Saloff-Coste, Convolution powers of complex functions on . Rev. Mat. Iberoam. 33 (2017), no. 3, pp. 1045–1121

DOI 10.4171/RMI/964