Convolution theorems associated with some integral operators and convolutions

Abstract

In this article, various convolution theorems involving certain weight functions and convolution products are derived. The convolution theorems we obtain are more general, convenient, and efficient than the complicated convolution theorem of the Hartley transform. Further results involving new variants of generalizations of Fourier and Hartley transforms are also discussed.

Some results of Watson and Plancherel‐type integral transforms related to the Hartley, Fourier convolutions and applications

“…Notably, the results for the Hartley and Fourier convolution involving certain weight functions was studied in $$$ {\mathbb{R}}^n $$$ (see Al‐Omari & Baleanu 25 ). The Hartley transform was proposed as an alternative to the Fourier transform by R. V. L. Hartley in 1942.…”

Extension of generalized Fox’s H-function operator to certain set of generalized integrable functions

“…A similar proof for this theorem can be easily deduced from [14][15][16]. Hence it has been omitted.…”

“…The field of generalized functions has been developed along the requirements of its applications in linear and nonlinear partial differential equations, geometry, mathematical physics, stochastic analysis as well as in harmonic analysis, both in theoretical and numerical aspects. The recent space of generalized functions or Boehmians is obtained by abstract algebra similar to that of field of quotients (see, e.g., [4,[9][10][11][12][13][14][15][16][17][18]). In this section, we investigate the Boehmian spaces where the Inayat integral operator is well defined.…”