Convolutions with the Continuous Primitive Integral

Abstract EN

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‖ℬ?, 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.