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

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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### References:

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