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. Underthe Alexiewicz norm, the space of integrable distributions is a Banach space. We define theconvolution 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.
"Convolutions with the Continuous Primitive Integral." Abstr. Appl. Anal. 2009 1 - 18, 2009. https://doi.org/10.1155/2009/307404