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.

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. Under the Alexiewicz norm, the space of integrable distributions is a Banach space. We define the convolution f∗g(x)=∫−∞∞f(x−y)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, f∗g is uniformly continuous and we have the estimate ‖f∗g‖∞≤‖f‖‖g‖ℬ𝒱, where ‖f‖=supI|∫If| is the Alexiewicz norm. This supremum is taken over all intervals I⊂ℝ. When g∈L1, the estimate is ‖f∗g‖≤‖f‖‖g‖1. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.