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Volume 13
Issue 15
10.3390/math13152435
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Horváth Spaces and a Representations of the Fourier Transform and Convolution

by
Emilio R. Negrín
1,2,
Benito J. González
1,2 and
Jeetendrasingh Maan
3,*
1
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de La Laguna (ULL), Campus de Anchieta, ES-38271 La Laguna, Spain
2
Instituto de Matemáticas y Aplicaciones (IMAULL), Universidad de La Laguna (ULL), ULL Campus de Anchieta, ES-38271 La Laguna, Spain
3
Department of Mathematics and Scientific Computing, National Institute of Technology, Hamirpur 177005, India
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(15), 2435; https://doi.org/10.3390/math13152435
Submission received: 2 July 2025 / Revised: 25 July 2025 / Accepted: 26 July 2025 / Published: 28 July 2025
(This article belongs to the Special Issue Special Functions with Applications)
Versions Notes

Abstract

This paper explores the structural representation and Fourier analysis of elements in Horváth distribution spaces Sk′, for k<−n. We prove that any element in Sk′ can be expressed as a finite sum of derivatives of continuous L1(Rn)-functions acting on Schwartz test functions. This representation leads to an explicit expression for their distributional Fourier transform in terms of classical Fourier transforms. Additionally, we present a distributional representation for the convolution of two such elements, showing that the convolution is well-defined over S. These results deepen our understanding of non-tempered distributions and extend Fourier methods to a broader functional framework.
Keywords: classical Fourier transform; distributional Fourier transform; representation of distributions; Horváth spaces; convolution classical Fourier transform; distributional Fourier transform; representation of distributions; Horváth spaces; convolution

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MDPI and ACS Style

Negrín, E.R.; González, B.J.; Maan, J. Horváth Spaces and a Representations of the Fourier Transform and Convolution. Mathematics 2025, 13, 2435. https://doi.org/10.3390/math13152435

AMA Style

Negrín ER, González BJ, Maan J. Horváth Spaces and a Representations of the Fourier Transform and Convolution. Mathematics. 2025; 13(15):2435. https://doi.org/10.3390/math13152435

Chicago/Turabian Style

Negrín, Emilio R., Benito J. González, and Jeetendrasingh Maan. 2025. "Horváth Spaces and a Representations of the Fourier Transform and Convolution" Mathematics 13, no. 15: 2435. https://doi.org/10.3390/math13152435

APA Style

Negrín, E. R., González, B. J., & Maan, J. (2025). Horváth Spaces and a Representations of the Fourier Transform and Convolution. Mathematics, 13(15), 2435. https://doi.org/10.3390/math13152435

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