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

摘要

In this article, we study the Watson‐type integral transforms for the convolutions related to the Hartley and Fourier transformations. We establish necessary and sufficient conditions for these operators to be unitary in the space L2(ℝ)$$ {L}_2left(mathbb{R}right) $$ and get their inverse represented in the conjugate symmetric form. Furthermore, we also formulate the Plancherel‐type theorem for the aforementioned operators, proved a sequence of functions that converge to the original function in the norm of L2(ℝ)$$ {L}_2left(mathbb{R}right) $$, and further show the boundedness of these operators. Besides showing the obtained results, we demonstrate how to use it to solve the class of integro‐differential equations Barbashin type, the differential equations, and the system of differential equations. Numerical examples are given for illustrating the obtained results to ensure their validity and applicability.