Online premeans and their computation complexity

We extend some approach to a family of symmetric means (i.e. symmetric functions $\mathscr{M} \colon \bigcup_{n=1}^\infty I^n \to I$ with $\min\le \mathscr{M}\le \max$; $I$ is an interval). Namely, it is known that every symmetric mean can be written in a form $\mathscr{M}(x_1,\dots,x_n):=F(f(x_1)+\cdots+f(x_n))$, where $f \colon I \to G$ and $F \colon G \to I$ ($G$ 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.

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