Convolution powers of complex functions on Zd

• Evan Randles ^[1] ; Laurent Saloff-Coste ^[2]
  1. [1] Colby College

     Colby College

     City of Waterville, Estados Unidos

     
  2. [2] Cornell University

     Cornell University

     City of Ithaca, Estados Unidos

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 33, Nº 3, 2017 (Ejemplar dedicado a: Yves Meyer), págs. 1045-1121
• Idioma: inglés
• DOI: 10.4171/RMI/964
• Texto completo no disponible (Saber más ...)
• Resumen

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