Department of Mathematics and Statistics, University of the Fraser Valley, Abbotsford, BC, V2S 7M8, Canada
Academic Editor: H. Bevan Thompson
Copyright © 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.
If is a continuous function on the real line and is its distributional derivative, then the continuous primitive integral of distribution is . 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 for an integrable distribution and a function of bounded variation or an function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. For of bounded variation,
is uniformly continuous and we have the estimate , where is the Alexiewicz norm. This supremum is taken over all intervals
. When , the estimate is . There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.