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

Janusz Matkowski. "Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem." Colloquium Mathematicae 133.1 (2013): 35-49. <http://eudml.org/doc/283818>.

@article{JanuszMatkowski2013,
abstract = {A generalization of the weighted quasi-arithmetic mean generated by continuous and increasing (decreasing) functions $f₁,...,f_\{k\}:I → ℝ$, k ≥ 2, denoted by $A^\{[f₁,...,f_\{k\}]\}$, is considered. Some properties of $A^\{[f₁,...,f_\{k\}]\}$, 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 $f_\{j\}:I → ℝ$, j ∈ ℕ, a mean $A^\{[f₁,f₂,...]\}: ⋃_\{k=1\}^\{∞\} I^\{k\} → I$ 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.},
author = {Janusz Matkowski},
journal = {Colloquium Mathematicae},
keywords = {means; quasi-arithmetic means; comparability of means; associativity of means; Kolmogorov-Nagumo theorem},
language = {eng},
number = {1},
pages = {35-49},
title = {Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem},
url = {http://eudml.org/doc/283818},
volume = {133},
year = {2013},
}

TY - JOUR
AU - Janusz Matkowski
TI - Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem
JO - Colloquium Mathematicae
PY - 2013
VL - 133
IS - 1
SP - 35
EP - 49
AB - A generalization of the weighted quasi-arithmetic mean generated by continuous and increasing (decreasing) functions $f₁,...,f_{k}:I → ℝ$, k ≥ 2, denoted by $A^{[f₁,...,f_{k}]}$, is considered. Some properties of $A^{[f₁,...,f_{k}]}$, 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 $f_{j}:I → ℝ$, j ∈ ℕ, a mean $A^{[f₁,f₂,...]}: ⋃_{k=1}^{∞} I^{k} → I$ 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.
LA - eng
KW - means; quasi-arithmetic means; comparability of means; associativity of means; Kolmogorov-Nagumo theorem
UR - http://eudml.org/doc/283818
ER -