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2009 Convolutions with the Continuous Primitive Integral
Erik Talvila
Abstr. Appl. Anal. 2009: 1-18 (2009). DOI: 10.1155/2009/307404

Abstract

If F is a continuous function on the real line and f=F is its distributional derivative, then the continuous primitive integral of distribution f is abf=F(b)F(a). This integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals. Underthe Alexiewicz norm, the space of integrable distributions is a Banach space. We define theconvolution fg(x)=f(xy)g(y)dy for f an integrable distribution and g a function of bounded variation or an L1 function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. For g of bounded variation,fg is uniformly continuous and we have the estimate fgfg𝒱, where f=supI|If| is the Alexiewicz norm. This supremum is taken over all intervalsI. When gL1, the estimate is fgfg1. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.

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Erik Talvila. "Convolutions with the Continuous Primitive Integral." Abstr. Appl. Anal. 2009 1 - 18, 2009. https://doi.org/10.1155/2009/307404

Information

Published: 2009
First available in Project Euclid: 16 March 2010

zbMATH: 1192.46039
MathSciNet: MR2559282
Digital Object Identifier: 10.1155/2009/307404

Rights: Copyright © 2009 Hindawi

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