Convolution Operator With a Weight Function for Hartley Integral Transform and Its Applications

In this paper, we introduce a new convolution operator with the weight function for Hartley integral transform. Several algebraic properties of this convolution operator are presented. In application, we apply this convolution operator to solving a class of integral equation and system of integral equations of Toeplitz plus Hankel type. On the other hand, we study the Watson- type integral transform for this convolution operator. We establish necessary and sufficient conditions for these operators to be unitary in the space and get their inverse represented in the conjungate symetric form. Moreover, we also formulate the Plancherel type theorem for aforementioned operators, prove a sequence of functions that converges to original function in the norm of and further show the boundedness of these operator. Finally, we study a class of integro- differential equation of Barashin type.