Convolutions with the Continuous Primitive Integral

Talvila, Erik
January 2009
Abstract & Applied Analysis;2009, Special section p1
Academic Journal
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 ∫baf = 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) = ∫ 8 −8f(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∥8 ≤ ∥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.


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