Convolutions with the Continuous Primitive Integral

Abstract

IfFis a continuous function on the real line andf=F′is its distributional derivative, then the continuous primitive integral of distributionfis∫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 convolutionf∗g(x)=∫−∞∞f(x−y)g(y)dyforfan integrable distribution andga function of bounded variation or anL1function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. Forgof bounded variation,f∗gis uniformly continuous and we have the estimate‖f∗g‖∞≤‖f‖‖g‖ℬ𝒱, where‖f‖=supI|∫If|is the Alexiewicz norm. This supremum is taken over all intervalsI⊂ℝ. Wheng∈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.

The one-dimensional heat equation in the Alexiewicz norm

“…Properties of the convolution are proven for the continuous primitive integral in [23]. It is shown there by a limiting process that f * g also exists for g ∈ L 1 and that f * g ≤ f g 1 .…”

“…(h) Let −∞ < α < β < ∞. Using the Fubini theorem in the Appendix to [23] and integrating by parts, we have…”

“…Taking the supremum over all such intervals gives the first result. Integrate by parts and use the Fubini-Tonelli theorem in [23] to get…”

“…to see that the conditions of [23,Proposition A.3] are satisfied. We then have equality between (5.3) and (5.7), upon interchanging orders of integration in (5.7) and changing variables.…”

Integrals and Banach spaces for finite order distributions

“…Under convolution L 1 is a Banach algebra. Although convolution has been defined in A 1 c × L 1 in [27] it does not seem possible to define convolution in A 1 c × A 1 c . Convolutions can be defined for distributions but restrictions on the supports are generally imposed.…”