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

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(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, is 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.

Automated summary

The paper introduces the concept of polyconvolution for Fourier-cosine and Laplace integral operators and explores its applications. It investigates the structure of this polyconvolution operator and associated integral transforms. The paper establishes necessary and sufficient conditions for the operator to be an isometric isomorphism and provides its inverse in the conjugate symmetric form. It also shows the correlation between the existence of polyconvolution and weighted spaces, and obtains Young's type theorem and norm inequalities in weighted space. Additionally, the paper investigates the solvability of certain integral equations with the help of factorization identities of polyconvolution and provides illustrative examples of the obtained results.