Topology and regularity for generalized ultradistribution algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908520625995776 |
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| author | Pilipovic, Stevan Risteski, Dragana Scarpalezos, Dimitris Zigic, Milica |
| author_facet | Pilipovic, Stevan Risteski, Dragana Scarpalezos, Dimitris Zigic, Milica |
| contents | Compiling essential results for non-quasianalytic ultradistribution spaces and Colombeau versions of generalized ultradistribution algebras, we analyze strong $B$- and strong $R$-association of a generalized ultradistribution $[(f_\varepsilon)]$. The strong association of $[(f_\varepsilon)]$ to a Komatsu-type ultradistribution $T$, with additional assumption on regularity of $[(f_\varepsilon)]$ of Beurling, respectively, Roumieu type, implies that $T$ is an ultradifferentiable function of Beurling, Roumieu type, respectively. We demonstrate that, under suitable conditions on regularity, a weakly negligible net $(g_\varepsilon)_{\varepsilon\in(0,1)}$ (meaning that the net of complex numbers $(\int g_\varepsilonϕdx)_{\varepsilon\in(0,1)}$ is Beurling, respectively, Roumieu negligible for every ultradifferentiable function $ϕ$ in the corresponding test space), is a negligible net in the sense of generalized ultradistributions. Furthermore, we prove that a translation invariant generalized ultradistribution $g$ is equal to a generalized constant in both types of generalized ultradistribution algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07054 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topology and regularity for generalized ultradistribution algebras Pilipovic, Stevan Risteski, Dragana Scarpalezos, Dimitris Zigic, Milica Functional Analysis Compiling essential results for non-quasianalytic ultradistribution spaces and Colombeau versions of generalized ultradistribution algebras, we analyze strong $B$- and strong $R$-association of a generalized ultradistribution $[(f_\varepsilon)]$. The strong association of $[(f_\varepsilon)]$ to a Komatsu-type ultradistribution $T$, with additional assumption on regularity of $[(f_\varepsilon)]$ of Beurling, respectively, Roumieu type, implies that $T$ is an ultradifferentiable function of Beurling, Roumieu type, respectively. We demonstrate that, under suitable conditions on regularity, a weakly negligible net $(g_\varepsilon)_{\varepsilon\in(0,1)}$ (meaning that the net of complex numbers $(\int g_\varepsilonϕdx)_{\varepsilon\in(0,1)}$ is Beurling, respectively, Roumieu negligible for every ultradifferentiable function $ϕ$ in the corresponding test space), is a negligible net in the sense of generalized ultradistributions. Furthermore, we prove that a translation invariant generalized ultradistribution $g$ is equal to a generalized constant in both types of generalized ultradistribution algebras. |
| title | Topology and regularity for generalized ultradistribution algebras |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2411.07054 |