Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2406.12491 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917102580924416 |
|---|---|
| author | Pasteczka, Paweł |
| author_facet | Pasteczka, Paweł |
| contents | We define so-called residual means, which have a Taylor expansion of the form $M(x)=\bar x +\tfrac12 ξ_M(\bar x) \text{Var}(x)+o(\|x-\bar x\|^α)$ for some $α>2$ and a single-variable function $ξ_M$ ($\bar x$ stands for the arithmetic mean of the vector $x$), and show that all symmetric means which are three times continuously differentiable are residual. We also calculate the value of residuum for quasideviation means and a few subclasses of this family.
Later, we apply it to establish the limit of the sequence $\big(\frac{\text{Var}\ {\bf M}^{n+1}(x)}{(\text{Var}\ {\bf M}^n(x))^2}\big)_{n=1}^\infty$, where ${\bf M} \colon I^p\to I^p$ is a mean-type mapping consisting of $p$-variable residual means on an interval $I$, and $x \in I^p$ is a nonconstant vector. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12491 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the new smoothness class of means and its impact to mean-type mappings Pasteczka, Paweł Classical Analysis and ODEs We define so-called residual means, which have a Taylor expansion of the form $M(x)=\bar x +\tfrac12 ξ_M(\bar x) \text{Var}(x)+o(\|x-\bar x\|^α)$ for some $α>2$ and a single-variable function $ξ_M$ ($\bar x$ stands for the arithmetic mean of the vector $x$), and show that all symmetric means which are three times continuously differentiable are residual. We also calculate the value of residuum for quasideviation means and a few subclasses of this family. Later, we apply it to establish the limit of the sequence $\big(\frac{\text{Var}\ {\bf M}^{n+1}(x)}{(\text{Var}\ {\bf M}^n(x))^2}\big)_{n=1}^\infty$, where ${\bf M} \colon I^p\to I^p$ is a mean-type mapping consisting of $p$-variable residual means on an interval $I$, and $x \in I^p$ is a nonconstant vector. |
| title | On the new smoothness class of means and its impact to mean-type mappings |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2406.12491 |