Convolutions of General Integration Operators. Applications

Abstract

In this chapter we consider systematically the problem of finding convolutions for an arbitrary right inverse operator of the differentiation operator in various function spaces, in particular for continuous, for locally integrable and for locally holomorphic functions. The resulting convolutions are used for an algebraic approach to the problem of developing locally holomorphic functions in Dirichlet series on a given exponent system. Thus we get the Leontiev expansion. A new problem solved by this algebraic approach is the multiplier problem for the formal Leontiev expansions of locally holomorphic functions. The convolutional approach is extended to some analogons of the general integration operators: the right inverses of the backward shift operator and Gelfond-Leontiev integration operator.