An existence and uniqueness result about algebras of Schwartz distributions
| Autor: | Dias, Nuno Costa, Jorge, Cristina, Prata, Joao Nuno |
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| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Monatshefte f\"ur Mathematik (2024) 203:43-61 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s00605-023-01917-z |
| Popis: | 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'impossibilit\'e 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 $\A$. Comment: 17 pages, to appear in Monatshefte f\"ur Mathematik |
| Databáze: | arXiv |
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