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Convolution powers of complex functions on Z
Author(s) -
Diaconis Persi,
SaloffCoste Laurent
Publication year - 2014
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200163
Subject(s) - mathematics , convolution (computer science) , smoothing , hermitian matrix , diagonal , limit (mathematics) , pure mathematics , markov chain , convolution power , fourier transform , convolution theorem , mathematical analysis , fourier analysis , geometry , statistics , artificial neural network , computer science , fractional fourier transform , machine learning
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 Thomée which are motivated by applications to data smoothing (Schoenberg and Greville) and finite difference schemes (Thomée). 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 k n ofM k = I − ( I − M ) kis obtained. This generalizes the Carne–Varopoulos Markov chain estimate.