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Abstract and Applied Analysis
Volume 2009, Article ID 307404, 18 pages
http://dx.doi.org/10.1155/2009/307404
Research Article

Convolutions with the Continuous Primitive Integral

Department of Mathematics and Statistics, University of the Fraser Valley, Abbotsford, BC, Canada V2S 7M8

Received 13 May 2009; Accepted 7 September 2009

Academic Editor: H. Bevan Thompson

Copyright Β© 2009 Erik Talvila. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

If 𝐹 is a continuous function on the real line and 𝑓=πΉξ…ž is its distributional derivative, then the continuous primitive integral of distribution 𝑓 is βˆ«π‘π‘Žπ‘“=𝐹(𝑏)βˆ’πΉ(π‘Ž). This integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals. Under the Alexiewicz norm, the space of integrable distributions is a Banach space. We define the convolution βˆ«π‘“βˆ—π‘”(π‘₯)=βˆžβˆ’βˆžπ‘“(π‘₯βˆ’π‘¦)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, π‘“βˆ—π‘” is uniformly continuous and we have the estimate β€–π‘“βˆ—π‘”β€–βˆžβ‰€β€–π‘“β€–β€–π‘”β€–β„¬π’±, where ‖𝑓‖=sup𝐼|βˆ«πΌπ‘“| is the Alexiewicz norm. This supremum is taken over all intervals πΌβŠ‚β„. When π‘”βˆˆπΏ1, the estimate is β€–π‘“βˆ—π‘”β€–β‰€β€–π‘“β€–β€–π‘”β€–1. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.