On the Novel Superpolyconvolution Operator for Six Integral Transforms Fourier Cosine, Fourier Sine, Laplace and Applications

Abstract

The main target of this article is to investigate the novel superpolyconvolution related to the six integral transforms Fourier cosine, Fourier sine and Laplace. The operator generalizes the classical polyconvolution framework by incorporating a six-fold transform decomposition that extends the traditional four-transform structure. Some interesting algebraic properties of this operator are presented. The Watson-type theorem is given. We also establish the Young-type inequalities in mixed \(L^p\) spaces, together with a Titchmarsh-type uniqueness theorem ensuring the distinctness of the associated functions. Applications are also discussed, showing that the proposed operator allows explicit solutions to various integral equations of Toeplitz plus Hankel type and to Barbashin-type equations.