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 $f={F}^{\prime}$ is its distributional derivative, then the continuous primitive integral of distribution $f$ is ${\int}_{a}^{b}f=F(b)-F(a)$. 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 $f\ast\,\!g(x)={\int}_{-\infty }^{\infty }f(x-y)g(y)dy$ for $f$ an integrable distribution and $g$ a function of bounded variation or an ${L}^{1}$ function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. For $g$ of bounded variation,$f\ast\,\!g$ is uniformly continuous and we have the estimate ${\Vert f\ast\,\!g\Vert }_{\infty }\leq \Vert f\Vert {\Vert g\Vert }_{\mathcal{B}\mathcal{V}}$, where $\Vert f\Vert ={\text{sup}}_{I}|{\int}_{I}f|$ is the Alexiewicz norm. This supremum is taken over all intervals$I\subset \mathbb{R}$. When $g\in {L}^{1}$, the estimate is $\Vert f\ast\,\!g\Vert \leq \Vert f\Vert {\Vert g\Vert }_{1}$. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.

## Citation

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