Convolutions with the continuous primitive integral. (English) Zbl 1192.46039

Talvila, Erik

Abstr. Appl. Anal. 2009, Article ID 307404, 18 p. (2009).

Summary: If \(F\) is a continuous function on the real line and \(f=F'\) is its distributional derivative, then the continuous primitive integral of the distribution \(f\) is \(\int_a^bf=F(b)-F(a)\). 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 \(f*g(x)=\int^\infty_{-\infty} f(x-y)g(y)\,dy\) 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 \(\|f*g\|_\infty\leq \|f\|\,\|g\|_{\mathcal{BV}}\), where \(\|f\|=\sup_I|\int_If|\) is the Alexiewicz norm. This supremum is taken over all intervals \(I\subset \mathbb R\). When \(g\in L^1\), the estimate is \(\|f*g\|\leq \|f\|\,\|g\|_1\). There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.

MSC:

46F10	Operations with distributions and generalized functions
28B05	Vector-valued set functions, measures and integrals
26A39	Denjoy and Perron integrals, other special integrals
44A35	Convolution as an integral transform

Keywords:

Alexiewicz norm; integrable distributions; convolution

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