New polyconvolution product for Fourier-cosine and Laplace integral operators and their applications

Abstract

The goal of this paper is to introduce the notion of polyconvolution for Fourier-cosine, Laplace integral operators, and its applications. The structure of this polyconvolution operator and associated integral transforms are investigated in detail. The Watson-type theorem is given, to establish necessary and sufficient conditions for this operator to be isometric isomorphism (unitary) on $L_2 (\mathbb{R}_+)$, and to get its inverse represented in the conjugate symmetric form. The correlation between the existence of polyconvolution with some weighted spaces is shown, and Young's type theorem, as well as the norm-inequalities in weighted space, are also obtained. As applications, we investigate the solvability of a class of Toeplitz plus Hankel type integral equations and linear Barbashin's equations with the help of factorization identities of such polyconvolution. Several examples are provided to illustrate the obtained results to ensure their validity and applicability.