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CONVOLUTION OPERATOR WITH A WEIGHT FUNCTION FOR HARTLEY INTEGRAL TRANSFORM AND ITS APPLICATIONS
Keywords:
Integral equation, convolution, generalized convolution, Hartley transformDOI:
https://doi.org/10.17654/0972087125036Abstract
In this paper, we introduce a new convolution operator with the weight function $\gamma(y)=\sin y$ 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 $L_2(\mathbb{R})$ 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 $L_2(\mathbb{R}),$ and further show the boundedness of these operator. Finally, we study a class of integro- differential equation of Barashin type.
Received: June 9, 2025
Revised: September 4, 2025
Accepted: September 12, 2025
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