Some results of Watson, Plancherel type integral transforms related to
the Hartley, Fourier convolutions and applications
In this work, 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 L 2 (R) space and get their inverse represented in the
conjugate symmetric form. Furthermore, we also formulated the
Plancherel-type theorem for the aforementioned operators and prove a
sequence of functions that converge to the original function in the
defined L 2 (R) norm. Next, we study the boundedness of the operators (T
k ). Besides, showing the obtained results, we demonstrate how to use it
to solve the class of integro-differential equations of Barbashin type,
the differential equations, and the system of differential equations.
And there are numerical examples given to illustrate these.