Horváth Spaces and a Representations of the Fourier Transform and Convolution

Emilio R. Negrín, Benito J. González and Jeetendrasingh Maan ()
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Emilio R. Negrín: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de La Laguna (ULL), Campus de Anchieta, ES-38271 La Laguna, Spain
Benito J. González: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de La Laguna (ULL), Campus de Anchieta, ES-38271 La Laguna, Spain
Jeetendrasingh Maan: Department of Mathematics and Scientific Computing, National Institute of Technology, Hamirpur 177005, India

Mathematics, 2025, vol. 13, issue 15, 1-10

Abstract: This paper explores the structural representation and Fourier analysis of elements in Horváth distribution spaces S k ′ , for k < − n . We prove that any element in S k ′ can be expressed as a finite sum of derivatives of continuous L 1 ( R n ) -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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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