Convolutions with the Continuous Primitive Integral - Erik Talvila (Abstract and Applied Analysis)
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.