Convolution powers of complex functions on Z

Repeated convolution of a probability measure on Z leads to the central limit theorem and other limit theorems. This paper investigates what kinds of results remain without positivity. It reviews theorems due to Schoenberg, Greville, and Thomee which are motivated by applications to data smoothing (Schoenberg and Greville) and finite difference schemes (Thomee). Using Fourier transform arguments, we prove detailed decay bounds for convolution powers of finitely supported complex functions on Z. If M is an hermitian contraction, an estimate for the off-diagonal entries of the powers M^n k of M^k = I -(I-M)^k is obtained. This generalizes the Carne-Varopoulos Markov chain estimate.