Multivariable generalizations of bivariate means via invariance
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911772151119872 |
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| author | Pasteczka, Paweł |
| author_facet | Pasteczka, Paweł |
| contents | For a given $p$-variable mean $M \colon I^p \to I$ ($I$ is a subinterval of $\mathbb{R}$), following (Horwitz, 2002) and (Lawson and Lim, 2008), we can define (under certain assumption) its $(p+1)$-variable $β$-invariant extension as the unique solution $K \colon I^{p+1} \to I$ of the functional equation \begin{align*} K\big(M(x_2,\dots,x_{p+1})&,M(x_1,x_3,\dots,x_{p+1}),\dots,M(x_1,\dots,x_p)\big)\\ &=K(x_1,\dots,x_{p+1}), \text{ for all }x_1,\dots,x_{p+1} \in I \end{align*} in the family of means.
Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04121 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multivariable generalizations of bivariate means via invariance Pasteczka, Paweł Dynamical Systems For a given $p$-variable mean $M \colon I^p \to I$ ($I$ is a subinterval of $\mathbb{R}$), following (Horwitz, 2002) and (Lawson and Lim, 2008), we can define (under certain assumption) its $(p+1)$-variable $β$-invariant extension as the unique solution $K \colon I^{p+1} \to I$ of the functional equation \begin{align*} K\big(M(x_2,\dots,x_{p+1})&,M(x_1,x_3,\dots,x_{p+1}),\dots,M(x_1,\dots,x_p)\big)\\ &=K(x_1,\dots,x_{p+1}), \text{ for all }x_1,\dots,x_{p+1} \in I \end{align*} in the family of means. Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions. |
| title | Multivariable generalizations of bivariate means via invariance |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2402.04121 |