Weighting by iteration: iterations of $$n$$ variables means based on subdivisions of the standard $$(n-1)$$-simplex.(English)Zbl 1481.26027

The author parametrizes means with the $$(n-1)$$ simplex $$\Delta_{n-1}=\{(w_1,w_2,\ldots,w_n)\in\mathbb{R}^n :w_i\geq 0, \sum_1^n w_i=1\}$$. After an introduction of some concepts from simplicial homology and iterated systems of functions (termed ISF), the author considers lower means and studies their properties. Following this comes the barycentric iteration of means via a two step algorithm by first making an assignment on the face of each $$\Delta_{n-1}$$ and iteratively applying the mean from previous assignments via $$M^{\frac{v_1+\cdots+v_n}{n}}(x)=M(M^{v_1}(x),\ldots,M^{v_n}(x))$$ and studies the properties of these iterated means. It is shown that for symmetric means the orientation of the simplex does not matter. Next, weighting procedures in the iteration as well as weighting based on other subdivision in the manner of H. Freudenthal [Ann. Math. (2) 43, 580–582 (1942; Zbl 0060.40701)] and properties of extension of the means are studied. The results are compared with those studied previously by other authors, in particular it is shown when the new methods coincide with J. Aczél’s [Bull. Am. Math. Soc. 54, 392–400 (1948; Zbl 0030.02702)] weighting procedure for means and turn out to be continuous and scale invariant.

MSC:

 26E60 Means 39B12 Iteration theory, iterative and composite equations 54C20 Extension of maps 54C30 Real-valued functions in general topology

Citations:

Zbl 0060.40701; Zbl 0030.02702
Full Text:

References:

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