Online premeans and their computation complexity

Abstract

We extend some approach to a family of symmetric means (i.e. symmetric functions $\mathcal{M}:{\bigcup }_{n=1}^{\infty }{I}^n\to I$ with $\min \le \mathcal{M}\le \max$; $I$ is an interval). Namely, it is known that every symmetric mean can be written in a form $\mathcal{M}(x_1,\dots ,x_n):=F(f(x_1)+\cdots +f(x_n))$, where $f:I\to G$ and $F:G\to I$ ($G$ is a commutative semigroup). For $G={\mathbb{R}}^k$ or $G={\mathbb{R}}^k\times \mathbb{Z}$ ($k\in \mathbb{N}$) and continuous functions $f$ and $F$ we obtain two series of families (depending on $k$). It can be treated as a measure of complexity in a family of means (this idea is inspired by theory of regular languages and algorithmics). As a result we characterize celebrated families of quasi-arithmetic means ($G=\mathbb{R}\times \mathbb{Z}$) and Bajraktarević means ($G={\mathbb{R}}^2$ under some additional assumptions). Moreover, we establish certain estimations of complexity for several other classical families.

References in corpus (1)

• Characterization of the Hardy property of means and the best Hardy constants