Section method and Frechet polynomials
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910608033579008 |
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| author | Daianu, Dan M |
| author_facet | Daianu, Dan M |
| contents | Using the section method we characterize the solutions $ f:U\rightarrow Y$ of the following four equations \begin{equation*} \sum\limits_{i=0}^{n}\left( -1\right) ^{n-i}\tbinom{n}{i}f\left( \sqrt[m]{ u^{m}+iv^{m}}\right) =\left( n!\right) f\left( v\right) \text{, } \end{equation*} \begin{equation*} f\left( u\right) +\sum\limits_{i=1}^{n+1}\left( -1\right) ^{i} \tbinom{n+1}{i}f\left( \sqrt[m]{u^{m}+iv^{m}}\right) =0, \end{equation*} \begin{equation*} \sum\limits_{i=0}^{n}\left( -1\right) ^{n-i}\tbinom{n}{i}f\left( \arcsin \left\vert \sin u\sin ^{i}v\right\vert \right) =\left( n!\right) f\left( v\right) \text{ and } \end{equation*} \begin{equation*} f\left( u\right) +\sum\limits_{i=1}^{n+1}\left( -1\right) ^{i}\tbinom{n+1}{i% }f\left( \arcsin \left\vert \sin u\sin ^{i}v\right\vert \right) =0, \end{equation*}
where $m\geq 2$ and $n$ are positive integers,$ \ U\subseteq
\mathbb{R}
$ is a maximally relevant real domain and $\left( Y,+\right) $ is an $\left( n!\right) $ -divisible Abelian group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11204 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Section method and Frechet polynomials Daianu, Dan M Analysis of PDEs Functional Analysis 39B52 (Primary) 39A70 (Secondary) Using the section method we characterize the solutions $ f:U\rightarrow Y$ of the following four equations \begin{equation*} \sum\limits_{i=0}^{n}\left( -1\right) ^{n-i}\tbinom{n}{i}f\left( \sqrt[m]{ u^{m}+iv^{m}}\right) =\left( n!\right) f\left( v\right) \text{, } \end{equation*} \begin{equation*} f\left( u\right) +\sum\limits_{i=1}^{n+1}\left( -1\right) ^{i} \tbinom{n+1}{i}f\left( \sqrt[m]{u^{m}+iv^{m}}\right) =0, \end{equation*} \begin{equation*} \sum\limits_{i=0}^{n}\left( -1\right) ^{n-i}\tbinom{n}{i}f\left( \arcsin \left\vert \sin u\sin ^{i}v\right\vert \right) =\left( n!\right) f\left( v\right) \text{ and } \end{equation*} \begin{equation*} f\left( u\right) +\sum\limits_{i=1}^{n+1}\left( -1\right) ^{i}\tbinom{n+1}{i% }f\left( \arcsin \left\vert \sin u\sin ^{i}v\right\vert \right) =0, \end{equation*} where $m\geq 2$ and $n$ are positive integers,$ \ U\subseteq \mathbb{R} $ is a maximally relevant real domain and $\left( Y,+\right) $ is an $\left( n!\right) $ -divisible Abelian group. |
| title | Section method and Frechet polynomials |
| topic | Analysis of PDEs Functional Analysis 39B52 (Primary) 39A70 (Secondary) |
| url | https://arxiv.org/abs/2409.11204 |