Convolutions with the Continuous Primitive Integral

Erik Talvila

doi:10.1155/2009/307404

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‖ℬ&#x1D4B1;, 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.

Highlights

Introduction and NotationThe convolution of two functions f and g on the real line is f ∗ g x f x−y g y dy.Convolutions play an important role in pure and applied mathematics in Fourier analysis, approximation theory, differential equations, integral equations, and many other areas
We extend the convolution f ∗ g to f ∈ AC and g ∈ L1
Let AC R be the functions that are absolutely continuous on each compact interval and which are of bounded variation on the real line

Summary

Introduction

Introduction and NotationThe convolution of two functions f and g on the real line is f ∗ g x f x−y g y dy.Convolutions play an important role in pure and applied mathematics in Fourier analysis, approximation theory, differential equations, integral equations, and many other areas. Denjoy integrals since their primitives are continuous functions. Each function of essential bounded variation has a distributional derivative that is a signed Radon measure. Let g χ{0}, g BV 2 but integration by parts shows f ∗ g 0 for each f ∈ AC.

Results

Conclusion