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Limit properties in a family of quasi-arithmetic means
- Prelegent(ci)
- Paweł Pasteczka
- Afiliacja
- Uniwersytet Warszawski
- Termin
- 24 października 2014 10:15
- Pokój
- p. 5840
- Seminarium
- Seminarium Zakładu Układów Dynamicznych
Quasi-arithmetic means directly generalize Power Means; they have been introduced in the works of Knopp, Kolmogorov and others. We will work towards the complete characterization of the property
$$\lim_{n \rightarrow \infty} f_n^{-1} \left( \frac{f_n(v_1)+\cdots+f_n(v_k)}{k} \right) \textrm{ for any }v \in I^k \textrm{ and } k \in \mathbb{N},$$
where $I$ is an interval, $(f_n)_{n \in \mathbb{N}}$ is a family of continuous, strictly monotone functions $f_n \colon I \rightarrow \mathbb{R}$. The crucial tool is the operator $f \mapsto f''/f'$ considered by Mikusi\'nski in the late 1940s. (This operator finds also applications in dynamical systems and mathematics of finance.)
$$\lim_{n \rightarrow \infty} f_n^{-1} \left( \frac{f_n(v_1)+\cdots+f_n(v_k)}{k} \right) \textrm{ for any }v \in I^k \textrm{ and } k \in \mathbb{N},$$
where $I$ is an interval, $(f_n)_{n \in \mathbb{N}}$ is a family of continuous, strictly monotone functions $f_n \colon I \rightarrow \mathbb{R}$. The crucial tool is the operator $f \mapsto f''/f'$ considered by Mikusi\'nski in the late 1940s. (This operator finds also applications in dynamical systems and mathematics of finance.)
