An existence and uniqueness result about algebras of Schwartz distributions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910450204016640 |
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| author | Dias, Nuno Costa Jorge, Cristina Prata, Joao Nuno |
| author_facet | Dias, Nuno Costa Jorge, Cristina Prata, Joao Nuno |
| contents | We prove that there exists essentially one {\it minimal} differential algebra of distributions $\A$, satisfying all the properties stated in the Schwartz impossibility result [L. Schwartz, Sur l'impossibilité de la multiplication des distributions, 1954], and such that $\C_p^{\infty} \subseteq \A \subseteq \DO' $ (where $\C_p^{\infty}$ is the set of piecewise smooth functions and $\DO'$ is the set of Schwartz distributions over $\RE$). This algebra is endowed with a multiplicative product of distributions, which is a generalization of the product defined in [N.C.Dias, J.N.Prata, A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients, 2009]. If the algebra is not minimal, but satisfies the previous conditions, is closed under anti-differentiation and the dual product by smooth functions, and the distributional product is continuous at zero then it is necessarily an extension of $\A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_14444 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An existence and uniqueness result about algebras of Schwartz distributions Dias, Nuno Costa Jorge, Cristina Prata, Joao Nuno Functional Analysis Classical Analysis and ODEs We prove that there exists essentially one {\it minimal} differential algebra of distributions $\A$, satisfying all the properties stated in the Schwartz impossibility result [L. Schwartz, Sur l'impossibilité de la multiplication des distributions, 1954], and such that $\C_p^{\infty} \subseteq \A \subseteq \DO' $ (where $\C_p^{\infty}$ is the set of piecewise smooth functions and $\DO'$ is the set of Schwartz distributions over $\RE$). This algebra is endowed with a multiplicative product of distributions, which is a generalization of the product defined in [N.C.Dias, J.N.Prata, A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients, 2009]. If the algebra is not minimal, but satisfies the previous conditions, is closed under anti-differentiation and the dual product by smooth functions, and the distributional product is continuous at zero then it is necessarily an extension of $\A$. |
| title | An existence and uniqueness result about algebras of Schwartz distributions |
| topic | Functional Analysis Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2309.14444 |