A direct and algebraic characterization of higher-order differential operators
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915241099526144 |
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| author | Fechner, Włodzimierz Gselmann, Eszter |
| author_facet | Fechner, Włodzimierz Gselmann, Eszter |
| contents | This paper presents an algebraic approach to characterizing higher-order differential operators. While the foundational Leibniz rule addresses first-order derivatives, its extension to higher orders typically involves identities relating multiple distinct operators. In contrast, we introduce a novel operator equation involving only a single $n$\textsuperscript{th}-order differential operator. We demonstrate that, under certain mild conditions, this equation serves to characterize such operators. Specifically, our results show that these higher-order differential operators can be identified as particular solutions to this single-operator identity. This approach provides a framework for understanding the algebraic structure of higher-order differential operators acting on function spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10100 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A direct and algebraic characterization of higher-order differential operators Fechner, Włodzimierz Gselmann, Eszter Classical Analysis and ODEs 47B92, 47J05, 47A62, 47B38, 39B22 This paper presents an algebraic approach to characterizing higher-order differential operators. While the foundational Leibniz rule addresses first-order derivatives, its extension to higher orders typically involves identities relating multiple distinct operators. In contrast, we introduce a novel operator equation involving only a single $n$\textsuperscript{th}-order differential operator. We demonstrate that, under certain mild conditions, this equation serves to characterize such operators. Specifically, our results show that these higher-order differential operators can be identified as particular solutions to this single-operator identity. This approach provides a framework for understanding the algebraic structure of higher-order differential operators acting on function spaces. |
| title | A direct and algebraic characterization of higher-order differential operators |
| topic | Classical Analysis and ODEs 47B92, 47J05, 47A62, 47B38, 39B22 |
| url | https://arxiv.org/abs/2504.10100 |