Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem

Janusz Matkowski

Colloquium Mathematicae, Tome 131 (2013), p. 35-49 / Harvested from The Polish Digital Mathematics Library

Résumé

A generalization of the weighted quasi-arithmetic mean generated by continuous and increasing (decreasing) functions $f ₁, . . ., f_{k} : I \to ℝ$ , k ≥ 2, denoted by $A^{[f ₁, . . ., f_{k}]}$ , is considered. Some properties of $A^{[f ₁, . . ., f_{k}]}$ , including “associativity” assumed in the Kolmogorov-Nagumo theorem, are shown. Convex and affine functions involving this type of means are considered. Invariance of a quasi-arithmetic mean with respect to a special mean-type mapping built of generalized means is applied in solving a functional equation. For a sequence of continuous strictly increasing functions $f_{j} : I \to ℝ$ , j ∈ ℕ, a mean $A^{[f ₁, f ₂, . . .]} : ⋃_{k = 1}^{\infty} I^{k} \to I$ is introduced and it is observed that, except symmetry, it satisfies all conditions of the Kolmogorov-Nagumo theorem. A problem concerning a generalization of this result is formulated.

Publié le : 2013-01-01

Zbl 1312.26055

EUDML-ID : urn:eudml:doc:283818

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     title = {Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem},
     journal = {Colloquium Mathematicae},
     volume = {131},
     year = {2013},
     pages = {35-49},
     zbl = {1312.26055},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-1-3}
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Janusz Matkowski. Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem. Colloquium Mathematicae, Tome 131 (2013) pp. 35-49. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-1-3/