Algebraic properties of the new polyconvolution operator related to the Laplace, Fourier cosine integral transforms and applications

Document Type : Research Article

Authors

Khoa Nguyen Minh

Le Trong Phuoc Bui

Van Thang Tran

Department of Mathematics, Electric Power University, 235 Hoang Quoc Viet, Nghia Do, Hanoi, Viet Nam.

10.22067/ijnao.2026.99197.1889

Abstract

This work is focused on investigating a new polyconvolution operator with the trigonometric weight function $\beta(y_1)=\cos y_1$ associated with the Laplace and Fourier cosine integral transforms. We give an explicit form of this operator, prove the corresponding factorization identity, and establish some important algebraic properties of it, together with a Titchmarsh-type theorem. Within this analytic setting, we obtain fundamental properties including $L_1$-boundedness, Young-type inequalities on mixed $L_p$ spaces, and a norm inequality of Saitoh's type in the $L_p(\Rp,q_i)$ weight spaces. Applications demonstrate the operator's effectiveness in solving some new classes of integral equations of Toeplitz plus Hankel type and of polyconvolution type, a system of equations of polyconvolution type, and an integro-differential equation of Barbashin type, yielding explicit closed-form solutions.

Keywords

Fourier cosine

Laplace integral transforms

Convolution

Generalized convolution

Polyconvolution

Integral equation

Integro-differential Barbashin equation

Subjects

• Numerical Analysis

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