Horváth Spaces and a Representations of the Fourier Transform and Convolution
This paper explores the structural representation and Fourier analysis of elements in Horváth distribution spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">...
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2025-07-01
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| author | Emilio R. Negrín Benito J. González Jeetendrasingh Maan |
| author_facet | Emilio R. Negrín Benito J. González Jeetendrasingh Maan |
| author_sort | Emilio R. Negrín |
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| description | This paper explores the structural representation and Fourier analysis of elements in Horváth distribution spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">S</mi><mi>k</mi><mo>′</mo></msubsup></semantics></math></inline-formula>, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo><</mo><mo>−</mo><mi>n</mi></mrow></semantics></math></inline-formula>. We prove that any element in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">S</mi><mi>k</mi><mo>′</mo></msubsup></semantics></math></inline-formula> can be expressed as a finite sum of derivatives of continuous <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>1</mn></msup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula>. These results deepen our understanding of non-tempered distributions and extend Fourier methods to a broader functional framework. |
| format | Article |
| id | doaj-art-4347c3a98ca54ab096bf2d2756b5b2e6 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-4347c3a98ca54ab096bf2d2756b5b2e62025-08-20T03:36:27ZengMDPI AGMathematics2227-73902025-07-011315243510.3390/math13152435Horváth Spaces and a Representations of the Fourier Transform and ConvolutionEmilio R. Negrín0Benito J. González1Jeetendrasingh Maan2Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de La Laguna (ULL), Campus de Anchieta, ES-38271 La Laguna, SpainDepartamento de Análisis Matemático, Facultad de Ciencias, Universidad de La Laguna (ULL), Campus de Anchieta, ES-38271 La Laguna, SpainDepartment of Mathematics and Scientific Computing, National Institute of Technology, Hamirpur 177005, IndiaThis paper explores the structural representation and Fourier analysis of elements in Horváth distribution spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">S</mi><mi>k</mi><mo>′</mo></msubsup></semantics></math></inline-formula>, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo><</mo><mo>−</mo><mi>n</mi></mrow></semantics></math></inline-formula>. We prove that any element in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">S</mi><mi>k</mi><mo>′</mo></msubsup></semantics></math></inline-formula> can be expressed as a finite sum of derivatives of continuous <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>1</mn></msup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula>. These results deepen our understanding of non-tempered distributions and extend Fourier methods to a broader functional framework.https://www.mdpi.com/2227-7390/13/15/2435classical Fourier transformdistributional Fourier transformrepresentation of distributionsHorváth spacesconvolution |
| spellingShingle | Emilio R. Negrín Benito J. González Jeetendrasingh Maan Horváth Spaces and a Representations of the Fourier Transform and Convolution Mathematics classical Fourier transform distributional Fourier transform representation of distributions Horváth spaces convolution |
| title | Horváth Spaces and a Representations of the Fourier Transform and Convolution |
| title_full | Horváth Spaces and a Representations of the Fourier Transform and Convolution |
| title_fullStr | Horváth Spaces and a Representations of the Fourier Transform and Convolution |
| title_full_unstemmed | Horváth Spaces and a Representations of the Fourier Transform and Convolution |
| title_short | Horváth Spaces and a Representations of the Fourier Transform and Convolution |
| title_sort | horvath spaces and a representations of the fourier transform and convolution |
| topic | classical Fourier transform distributional Fourier transform representation of distributions Horváth spaces convolution |
| url | https://www.mdpi.com/2227-7390/13/15/2435 |
| work_keys_str_mv | AT emiliornegrin horvathspacesandarepresentationsofthefouriertransformandconvolution AT benitojgonzalez horvathspacesandarepresentationsofthefouriertransformandconvolution AT jeetendrasinghmaan horvathspacesandarepresentationsofthefouriertransformandconvolution |