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

Abstract

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 andF . The open question about Hermitian weight-function of generalized convolution is posed at the end of the paper.

Solvability of some classes of singular integral equations of convolution type via Riemann–Hilbert problem

“…Meanwhile, we remark that the methods of this paper may be used to solving the above equations in the non-normal case. Indeed, it is possible to study the above-mentioned equation in Clifford analysis, which is similar to that in [24][25][26][27][28]. Further discussion is omitted here.…”

Convolutions and integral equations weighted by multi-dimensional Hermite functions

“…In addition, further functional characteristics of those convolutions will be investigated. We can see different applications of integral transforms and convolutions in [1,2,3,4,5,6,7,8,9,12,14,15,16].…”

The finite Hartley new convolutions and solvability of the integral equations with Toeplitz plus Hankel kernels

“…Feldman et al [13] provided a sufficient and necessary condition via partial indices for the invertibility of the Toeplitz integral operator (without Hankel term) in the space of Lebesgue integrable functions on finite intervals. In recent times, the papers [14][15][16][17][18] dealing with the cases of infinite domains of integration have been published.…”