Mellin and Widder–Lambert Transforms with Applications in the Salem Equivalence to the Riemann Hypothesis
This paper presents a comprehensive study of Plancherel’s theorem and inversion formulae for the Widder–Lambert transform, extending its scope to Lebesgue integrable functions, compactly supported distributions, and regular distributions with compact support. By employing the Plancherel theorem for...
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2025-02-01
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| author | Emilio R. Negrín Jeetendrasingh Maan Benito J. González |
| author_facet | Emilio R. Negrín Jeetendrasingh Maan Benito J. González |
| author_sort | Emilio R. Negrín |
| collection | DOAJ |
| description | This paper presents a comprehensive study of Plancherel’s theorem and inversion formulae for the Widder–Lambert transform, extending its scope to Lebesgue integrable functions, compactly supported distributions, and regular distributions with compact support. By employing the Plancherel theorem for the classical Mellin transform, we derive a corresponding Plancherel’s theorem specific to the Widder–Lambert transform. This novel approach highlights an intriguing connection between these integral transforms, offering new insights into their role in harmonic analysis. Additionally, we explore a class of functions that satisfy Salem’s equivalence to the Riemann hypothesis, providing a deeper understanding of the interplay between such equivalences and integral transforms. These findings open new avenues for further research on the Riemann hypothesis within the framework of integral transforms. |
| format | Article |
| id | doaj-art-5dff7c5f8a7e424f94e155a3b4c11c20 |
| institution | DOAJ |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-5dff7c5f8a7e424f94e155a3b4c11c202025-08-20T02:44:44ZengMDPI AGAxioms2075-16802025-02-0114212910.3390/axioms14020129Mellin and Widder–Lambert Transforms with Applications in the Salem Equivalence to the Riemann HypothesisEmilio R. Negrín0Jeetendrasingh Maan1Benito J. González2Departamento de Análisis Matemático, Universidad de La Laguna (ULL), ES-38271 La Laguna, Tenerife, SpainDepartment of Mathematics and Scientific Computing, National Institute of Technology, Hamirpur 177005, IndiaDepartamento de Análisis Matemático, Universidad de La Laguna (ULL), ES-38271 La Laguna, Tenerife, SpainThis paper presents a comprehensive study of Plancherel’s theorem and inversion formulae for the Widder–Lambert transform, extending its scope to Lebesgue integrable functions, compactly supported distributions, and regular distributions with compact support. By employing the Plancherel theorem for the classical Mellin transform, we derive a corresponding Plancherel’s theorem specific to the Widder–Lambert transform. This novel approach highlights an intriguing connection between these integral transforms, offering new insights into their role in harmonic analysis. Additionally, we explore a class of functions that satisfy Salem’s equivalence to the Riemann hypothesis, providing a deeper understanding of the interplay between such equivalences and integral transforms. These findings open new avenues for further research on the Riemann hypothesis within the framework of integral transforms.https://www.mdpi.com/2075-1680/14/2/129Mellin transformWidder–Lambert transformPlancherel’s theoreminversion formulaedistributions of compact supportSalem’s equivalence |
| spellingShingle | Emilio R. Negrín Jeetendrasingh Maan Benito J. González Mellin and Widder–Lambert Transforms with Applications in the Salem Equivalence to the Riemann Hypothesis Axioms Mellin transform Widder–Lambert transform Plancherel’s theorem inversion formulae distributions of compact support Salem’s equivalence |
| title | Mellin and Widder–Lambert Transforms with Applications in the Salem Equivalence to the Riemann Hypothesis |
| title_full | Mellin and Widder–Lambert Transforms with Applications in the Salem Equivalence to the Riemann Hypothesis |
| title_fullStr | Mellin and Widder–Lambert Transforms with Applications in the Salem Equivalence to the Riemann Hypothesis |
| title_full_unstemmed | Mellin and Widder–Lambert Transforms with Applications in the Salem Equivalence to the Riemann Hypothesis |
| title_short | Mellin and Widder–Lambert Transforms with Applications in the Salem Equivalence to the Riemann Hypothesis |
| title_sort | mellin and widder lambert transforms with applications in the salem equivalence to the riemann hypothesis |
| topic | Mellin transform Widder–Lambert transform Plancherel’s theorem inversion formulae distributions of compact support Salem’s equivalence |
| url | https://www.mdpi.com/2075-1680/14/2/129 |
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