A Novel Superpolyconvolution Operator and Applications

Nguyen, M. K.; Phuoc Bui, Le Trong

Indian Journal of Mathematics

A Novel Superpolyconvolution Operator and Applications

Author(s): M. K. Nguyen M. K. Nguyen Department of Mathematics
Electric Power University
235 Hoang Quoc Viet, Nghia Do, Ha Noi
Vietnam. khoanm@epu.edu.vn 0000-0001-7309-4456 , Le Trong Phuoc Bui Le Trong Phuoc Bui Department of Mathematics
Electric Power University
235 Hoang Quoc Viet, Nghia Do, Ha Noi
Vietnam. phuocblt@epu.edu.vn

Vol. 68, No. 1 (2026) • pp. 87–121

Published 05-11-2026 Article History Received 12-08-2026 Accepted 30-10-2026 Published 05-11-2026

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Abstract

We investigate a superpolyconvolution operator associated with the Fourier sine integral transform and characterized by the weight function $\gamma(y_1) = \sin(y_1)$. The operator generalizes the classical polyconvolution framework by incorporating a five-fold transform decomposition that extends the traditional four-transform structure. Within this analytic setting, we establish several fundamental results, including algebraic properties, $L_1$-integrability, a Parseval-type identity, and Young-type inequalities in mixed $L_p$ spaces, together with a Titchmarsh-type uniqueness theorem ensuring the distinctness of the associated functions. Applications are also discussed, showing that the proposed operator allows explicit solutions to various integral equations and integro-differential equation of Barbashin type.

Keywords

Trigonometric Fourier transforms; convolution; generalized convolution; polyconvolution; superpolyconvolution; Young's inequalities; integral equation; Barbashin's equation.

2020 Mathematics Subject Classification

42A38, 44A35, 45E10, 45J05, 26D10.

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Allahabad Mathematical Society

A Novel Superpolyconvolution Operator and Applications

Abstract

Keywords

2020 Mathematics Subject Classification

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