On the polyconvolution for the Fourier cosine, Fourier sine, and Kontorovich–Lebedev integral transforms

Abstract

The polyconvolution $ \mathop {*}\limits_1 \left( {f,g,h} \right)(x) $ of three functions f, g, and h is constructed for the Fourier cosine (F c ), Fourier sine (F s ), and Kontorovich–Lebedev (K iy ) integral transforms whose factorization equality has the form $$ {F_c}\left( {\mathop {*}\limits_1 \left( {f,g,h} \right)} \right)(y) = \left( {{F_s}f} \right)(y).\left( {{F_s}g} \right)(y).\left( {{K_{iy}}h} \right)\,\,\,\,\forall y > 0. $$ The relationships between this polyconvolution, the Fourier convolution, and the Fourier cosine convolution are established. In addition, we also establish the relationships between the product of the new polyconvolution and the products of the other known types of convolutions. As an application, we consider a class of integral equations with Toeplitz and Hankel kernels whose solutions can be obtained with the help of the new polyconvolution in the closed form. We also present the applications to the solution of systems of integral equations.