The solvability and explicit solutions of two integral equations via generalized convolutions

This paper presents the necessary and sufficient conditions for the solvability of two integral equations of convolution type; the first equation generalizes from integral equations with the Gaussian kernel, and the second one contains the Toeplitz plus Hankel kernels. Furthermore, the paper shows that the normed rings on L 1 ( R d ) are constructed by using the obtained convolutions, and an arbitrary Hermite function and appropriate linear combination of those functions are the weight-function of four generalized convolutions associating F and F ˇ . The open question about Hermitian weight-function of generalized convolution is posed at the end of the paper.