Abstract
We extend some approach to the family of symmetric means (i.e. symmetric functions mathscr {M} :bigcup _{n=1}^infty I^n rightarrow 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}(v_1,dots ,v_n):=F(f(v_1)+cdots +f(v_n)), where f :I rightarrow G and F :G rightarrow 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 the 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.
Highlights
In most cases means are defined using explicit formulas
For G = Rk or G = Rk × Z (k ∈ N) and continuous functions f and F we obtain two series of families. It can be treated as a measure of complexity in a family of means
As a result we characterize the celebrated families of quasi-arithmetic means (G = R × Z) and Bajraktarevic means (G = R2 under some additional assumptions)
Summary
In most cases means are defined using explicit formulas. There are only few general approaches to this topic. One of the most famous are so-called Chisini means (or level-surface means) [8] which allows to express all reflexive means in a unified form. We provide alternative way of defining means based on the idea emerging from the theory of regular languages. Our results bind two different scopes which, to the best of author’s knowledge, were not considered earlier. Due to this fact introduction is divided into few parts which are devoted to means
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