Convolution powers of complex functions on Z. Issue 10 (1st February 2014)

• Record Type: Journal Article
• Authors: Diaconis, Persi; Saloff‐Coste, Laurent
• Is Part Of: Mathematische Nachrichten. Volume 287:Issue 10(2014)
• Page Start: 1106
• Page End: 1130
• Publication Date: 2014-02-01
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201200163
• Languages: English
• ISSNs: 0025-584X

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>Repeated convolution of a probability measure on <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh1f8s6j1x" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:dummy:mana201200163:equation:mana201200163-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:math></alternatives></inline-formula> 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 <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh1f8s6hxr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:dummy:mana201200163:equation:mana201200163-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:math></alternatives></inline-formula>. If <italic>M</italic> is an hermitian contraction, an estimate for the off‐diagonal entries of the powers <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh1f8s6hw6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:dummy:mana201200163:equation:mana201200163-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>M</mml:mi><mml:mi>k</mml:mi><mml:mi>n</mml:mi></mml:msubsup></mml:math></alternatives></inline-formula> of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh1f8s6j0c" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:dummy:mana201200163:equation:mana201200163-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>I</mml:mi><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>I</mml:mi><mml:mo>−</mml:mo><mml:mi>M</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula> is obtained. This generalizes the Carne–Varopoulos Markov chain estimate.</p> </abstract>