Convolutions with the Continuous Primitive Integral

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 ${\Vert f\ast g\Vert }_{\infty }\le \Vert f\Vert {\Vert g\Vert }_{ℬ𝒱}$, where $\Vert f\Vert ={\text{sup}}_I\left|{\int }_{I}f\right|$ is the Alexiewicz norm. This supremum is taken over all intervals$I\subset ℝ$. When $g\in L^1$, the estimate is $\Vert f\ast g\Vert \le \Vert f\Vert {\Vert g\Vert }_1$. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.