Convolutions with the Continuous Primitive Integral

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 . 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 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 ∗ g is uniformly continuous and we have the estimate ∥ f ∗ g ∥ ∞ ≤ ∥ f ∥∥ g ∥ ℬ 𝒱 , where ∥ f ∥ = sup I |∫ I f | is the Alexiewicz norm. This supremum is taken over all intervals I ⊂ ℝ . When g ∈ L 1 , 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.

Locations

• DOAJ (DOAJ: Directory of Open Access Journals) - View
• arXiv (Cornell University) - View - PDF
• Project Euclid (Cornell University) - View - PDF
• Abstract and Applied Analysis - View - PDF