On the generalized convolution for F c , F s , and K–L integral transforms

Abstract

We study new generalized convolutions $$ f\mathop{*}\limits^{\gamma } g $$ with weight function γ(y) = y for the Fourier cosine, Fourier sine, and Kontorovich–Lebedev integral transforms in weighted function spaces with two parameters $$ L\left( {{\mathbb{R}_{{ + }}},{x^{\alpha }}{e^{{ - \beta x}}}dx} \right) $$ . These generalized convolutions satisfy the factorization equalities $$ {F_{{\left\{ {\begin{array}{*{20}{c}} s c \end{array} } \right\}}}}{\left( {f\mathop{*}\limits^{\gamma } g} \right)_{{\left\{ {\begin{array}{*{20}{c}} 1 2 \end{array} } \right\}}}}(y) = y\left( {{F_{{\left\{ {\begin{array}{*{20}{c}} c s \end{array} } \right\}}}}f} \right)(y)\left( {{K_{{iy}}}g} \right)\;\;\;\forall y > 0. $$ We establish a relationship between these generalized convolutions and known convolutions, and also relations that associate them with other convolution operators. As an example, we use these new generalized convolutions for the solution of a class of integral equations with Toeplitz-plus-Hankel kernels and a class of systems of two integral equations with Toeplitz-plus-Hankel kernels.