New convolutions and their applicability to integral equations of Wiener‐Hopf plus Hankel type

Abstract

We propose four new convolutions exhibiting convenient factorization properties associated with two finite interval integral transformations of Fourier‐type together with their norm inequalities. Moreover, we study the solvability of a class of integral equations of Wiener‐Hopf plus Hankel type (on finite intervals) with the help of the factorization identities of such convolutions. Fourier‐type series are used to produce the solution formula of such equations, and a Shannon‐type sampling formula is also obtained.

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.…”

A new convolution operator for the linear canonical transform with applications

“…Convolutions and the so-called "convolution type operators" (Bogveradze and Castro, 2008, Castro and Saitoh, 2012, Castro and Speck, 2000, Castro et al, 2020 are very important mathematical objects which are used in the modeling of a great diversity of applied problems. As it is well known, the classical convolution operator " * " (Bracewell and Bracewell, 1986) is given by:…”