On the Polyconvolution with the Weight Function for the Fourier Cosine, Fourier Sine, and the Kontorovich-Lebedev Integral Transforms

The polyconvolution with the weight function ð ›¾ of three functions ð ‘“ , ð ‘” , and â„Ž for the integral transforms Fourier sine ( ð ¹ ð ‘ ) , Fourier cosine ( ð ¹ ð ‘ ) , and Kontorovich-Lebedev ( ð ¾ ð ‘– 𠑦 ) , which is denoted by ð ›¾ ∗ ( ð ‘“ , ð ‘” , â„Ž ) (x) , has been constructed. This polyconvolution satisfies the following factorization property ð ¹ ð ‘ ( ð ›¾ ∗ ( ð ‘“ , ð ‘” , â„Ž ) ) ( 𠑦 ) = s i n 0 ð ‘¥ 0 0 0 ð ‘Ž 0 𠑦 ( ð ¹ ð ‘ ð ‘“ ) ( 𠑦 ) â‹… ( ð ¹ ð ‘ ð ‘” ) ( 𠑦 ) â‹… ( ð ¾ ð ‘– 𠑦 â„Ž ) ( 𠑦 ) , for all 𠑦 > 0 . The relation of this polyconvolution to the Fourier convolution and the Fourier cosine convolution has been obtained. Also, the relations between the polyconvolution product and others convolution product have been established. In application, we consider a class of integral equations with Toeplitz plus Hankel kernel whose solution in closed form can be obtained with the help of the new polyconvolution. An application on solving systems of integral equations is also obtained.