Convolutions of Generating Operators

Abstract

We considered convolutions of the generalized H-transforms in Chapter 11. The main property of these convolutions is the following {fy205-1} where (H _a f)(x) is the generalized H-transform with the power weight (11.1). It follows from this relation that the H-convolution (f ^a_* )(x) is connected with some integral transform and this connection is reflected in the names of other convolutions (Laplace convolution, Mellin convolution, et c.). A different approach to the definition of convolution, which connects some other operator with a convolution has been proposed by I.H. Dimovski ((1966)–(1981)). His definition is more suitable in developing a Mikusinski type operational calculus. In this chapter we will deal with convolutions in the Dimovski’s sense.