A refined variant of Hartley convolution: algebraic structures, spectral radius and related issues
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908838614007808 |
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| author | Tuan, Trinh |
| author_facet | Tuan, Trinh |
| contents | In this work, we propose a novel convolution product associated with the $\mathscr{H}$-transform, denoted by $\underset{\mathscr{H}}{\ast}$, and explore its fundamental properties. Here, the $\mathscr{H}$-transform may be regarded as a refined variant of the classical Fourier, Hartley transform, with kernel function depending on two parameters $a,b$. Our first contribution shows that the space of integrable functions, equipped with multiplication given by the $\underset{\mathscr{H}}{\ast}$-convolution, constitutes the commutative Banach algebra over the complex field, albeit without an identity element. Second, establishes the Wiener--Lévy type invertibility criterion for $\mathscr{H}$-algebras, obtained through the density property and process of unitarization, which serves as a key step toward the proof of Gelfand's spectral radius theorem. Third, provides an explicit upper-bound of Young's inequality for $\underset{\mathscr{H}}{\ast}$-convolution and its direct corollary. Finally, all of these theoretical findings are applied to analyze specific classes of the Fredholm integral equations and heat source problems, yielding a priori estimates under the established assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17529 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A refined variant of Hartley convolution: algebraic structures, spectral radius and related issues Tuan, Trinh Functional Analysis Classical Analysis and ODEs 46J10, 47A10, 44A35, 47A30 In this work, we propose a novel convolution product associated with the $\mathscr{H}$-transform, denoted by $\underset{\mathscr{H}}{\ast}$, and explore its fundamental properties. Here, the $\mathscr{H}$-transform may be regarded as a refined variant of the classical Fourier, Hartley transform, with kernel function depending on two parameters $a,b$. Our first contribution shows that the space of integrable functions, equipped with multiplication given by the $\underset{\mathscr{H}}{\ast}$-convolution, constitutes the commutative Banach algebra over the complex field, albeit without an identity element. Second, establishes the Wiener--Lévy type invertibility criterion for $\mathscr{H}$-algebras, obtained through the density property and process of unitarization, which serves as a key step toward the proof of Gelfand's spectral radius theorem. Third, provides an explicit upper-bound of Young's inequality for $\underset{\mathscr{H}}{\ast}$-convolution and its direct corollary. Finally, all of these theoretical findings are applied to analyze specific classes of the Fredholm integral equations and heat source problems, yielding a priori estimates under the established assumptions. |
| title | A refined variant of Hartley convolution: algebraic structures, spectral radius and related issues |
| topic | Functional Analysis Classical Analysis and ODEs 46J10, 47A10, 44A35, 47A30 |
| url | https://arxiv.org/abs/2509.17529 |