Convolutions of Linear Operators. Multipliers and Multiplier Quotients

Abstract

The Duhamel convolution $$(f * g)(t) = \int\limits_0^t f (t - \tau )d\tau $$ (1) is of considerable importance in many problems of analysis and its applications. It had been put by Mikusinski [77] as a basis of his direct approach to the Heaviside operational calculus. This approach is based on the connection between the Volterra integration operator $$lf(t) = \int\limits_0^t f (\tau )d\tau $$ (2) and the Duhamel convolution. This connection is expressed saying that l is the convolutional operator {1}*, i. e. lf={1}*f. This relation allows to obtain explicit representations of the commutants of Volterra’s integration operator in various function spaces. The considerations in the present section are a preliminary but typical illustration of the general convolutional scheme developed in the next sections.