The Sneddon R-Transform and Its Inverse over Lebesgue Spaces

Srivastava, Hari Mohan; Negrín, Emilio R.; Maan, Jeetendrasingh

doi:10.3390/axioms15010063

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The Sneddon R-Transform and Its Inverse over Lebesgue Spaces

by

Hari Mohan Srivastava

^1,2,3,4,5,6

,

Emilio R. Negrín

^7,8,*

and

Jeetendrasingh Maan

^9,*

¹

Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada

²

Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

³

Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea

⁴

Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taoyuan City 320314, Taiwan

⁵

Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan

⁶

Section of Mathematics, International Telematic University Uninettuno, 39 Corso Vittorio Emanuele II, I-00186 Rome, Italy

⁷

Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de La Laguna (ULL), Campus de Anchieta, ES-38271 La Laguna, Spain

⁸

Instituto de Matemáticas y Aplicaciones (IMAULL), Universidad de La Laguna (ULL), ULL Campus de Anchieta, ES-38271 La Laguna, Spain

⁹

Department of Mathematics and Scientific Computing, National Institute of Technology, Hamirpur 177005, India

^*

Authors to whom correspondence should be addressed.

Axioms 2026, 15(1), 63; https://doi.org/10.3390/axioms15010063

Submission received: 10 December 2025 / Revised: 9 January 2026 / Accepted: 11 January 2026 / Published: 16 January 2026

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Abstract

We study the Sneddon

R

-transform and its inverse in the setting of Lebesgue spaces. Generated by the mixed trigonometric kernel

x cos (x t) + h sin (x t)

, the

R

-transform acts as a unifying operator for sine- and cosine-type integral transforms. Boundedness, continuity, and weighted

L^{p}

-estimates are established in an appropriate Banach space framework, together with Parseval–Goldstein type identities. Initial and final value theorems are derived for generalized functions in Zemanian-type spaces, yielding precise asymptotic behaviour at the origin and at infinity. A finite-interval theory is also developed, leading to polynomial growth estimates and final value theorems for the finite

R

-transform.

Keywords: Sneddon R-transform; inverse transform; Lebesgue spaces; Parseval–Goldstein identity; weighted Lp-boundedness; generalized functions; finite R-transform

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MDPI and ACS Style

Srivastava, H.M.; Negrín, E.R.; Maan, J. The Sneddon R-Transform and Its Inverse over Lebesgue Spaces. Axioms 2026, 15, 63. https://doi.org/10.3390/axioms15010063

AMA Style

Srivastava HM, Negrín ER, Maan J. The Sneddon R-Transform and Its Inverse over Lebesgue Spaces. Axioms. 2026; 15(1):63. https://doi.org/10.3390/axioms15010063

Chicago/Turabian Style

Srivastava, Hari Mohan, Emilio R. Negrín, and Jeetendrasingh Maan. 2026. "The Sneddon R-Transform and Its Inverse over Lebesgue Spaces" Axioms 15, no. 1: 63. https://doi.org/10.3390/axioms15010063

APA Style

Srivastava, H. M., Negrín, E. R., & Maan, J. (2026). The Sneddon R-Transform and Its Inverse over Lebesgue Spaces. Axioms, 15(1), 63. https://doi.org/10.3390/axioms15010063

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