Convolution theorems related with the solvability of Wiener-Hopf plus Hankel integral equations and Shannon’s sampling formula

Abstract

This paper considers two finite integral transforms of Fourier-type, in view to propose a set of eight new convolutions, and to analyze the solvability of a class of the integral equations of Wiener-Hopf plus Hankel type, defined on finite intervals, which is involved in engineering problems. The solvability and solution of the considered equations are investigated by means of Fourier-type series, and a Shannon-type sampling formula is obtained. Some concluding remarks with respect to theoretical issues and engineering applications are emphasized in the last section, along with the analysis of some illustrative cases, which exemplify that the present method solves cases which are not under the conditions of previously known techniques.

New polyconvolution product for Fourier‐cosine and Laplace integral operators and their applications

“…We show a unique explicit solution on L1(R+) via Winner-Lévy's theorem. These are in stark contrast to recent results of Castro et al in [37,38] when the kernel K(x, y) is generated by two discriminant functions p (Hankel kernel) and q (Wiener-Hopf kernel) are 2π-periodic defined on one finite interval and the solution is obtained on the Hilbert space L2([0, 2π]) based on Shannon's sampling method.…”

“…We show a unique explicit solution on $$$ {L}_1\left({\mathrm{\mathbb{R}}}_{+}\right) $$$ via Winner–Lévy's theorem. These are in stark contrast to recent results of Castro et al in [37, 38] when the kernel $$$ K\left(x,y\right) $$$ is generated by two discriminant functions $$$ p $$$ (Hankel kernel) and $$$ q $$$ (Wiener–Hopf kernel) are $$$ 2\pi $$$‐periodic defined on one finite interval and the solution is obtained on the Hilbert space $$$ {L}_2\left(\left[0,2\pi \right]\right) $$$ based on Shannon's sampling method.…”

New Convolutions with Hermite Weight Functions

“…Despite all the previous developments in the area of convolution type operators and equations, and their applications (cf., e.g., [6]), several additional investigations in this field are going on in recent years. Namely, new convolutions are continuing to be introduced and applied to a great number of situations with particular emphasis within engineering problems (cf., e.g., [1,2,3,4,5]). Anyway, the majority of the introduced new convolutions are being considered in the full real line (or "full space") situation in which the use of the full space facilitates the action of the convolutions (as well as the existence and representation of their inverses).…”