Convolutions with the Continuous Primitive Integral

Erik Talvila

Abstract and Applied Analysis, 2009, vol. 2009, issue 1

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.

Date: 2009
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https://doi.org/10.1155/2009/307404

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2009:y:2009:i:1:n:307404

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