TITLE

# Convolutions with the Continuous Primitive Integral

AUTHOR(S)
Talvila, Erik
PUB. DATE
January 2009
SOURCE
Abstract & Applied Analysis;2009, Special section p1
SOURCE TYPE
DOC. TYPE
Article
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 âˆ«baf = F(b) âˆ’ 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 âˆ— g(x) = âˆ« 8 âˆ’8f(x âˆ’ 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 âˆ— g is uniformly continuous and we have the estimate âˆ¥f âˆ— gâˆ¥8 â‰¤ âˆ¥fâˆ¥âˆ¥gâˆ¥BV, where âˆ¥fâˆ¥ = supI âˆ£âˆ«Ifâˆ£ is the Alexiewicz norm. This supremum is taken over all intervals I âŠ‚ R. When g âˆˆ L1, the estimate is âˆ¥f âˆ— gâˆ¥ â‰¤ âˆ¥fâˆ¥âˆ¥gâˆ¥ 1. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.
ACCESSION #
55252793

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