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arXiv · 2402.04121

Multivariable generalizations of bivariate means via invariance

Paweł Pasteczka
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Abstract

For a given $p$-variable mean $M \colon I^p \to I$ ($I$ is a subinterval of $\mathbb{R}$), following (Horwitz, 2002) and (Lawson and Lim, 2008), we can define (under certain assumption) its $(p+1)$-variable $\beta$-invariant extension as the unique solution $K \colon I^{p+1} \to I$ of the functional equation \begin{align*} K\big(M(x_2,\dots,x_{p+1})&,M(x_1,x_3,\dots,x_{p+1}),\dots,M(x_1,\dots,x_p)\big)\\ &=K(x_1,\dots,x_{p+1}), \text{ for all }x_1,\dots,x_{p+1} \in I \end{align*} in the family of means. Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions.

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Paweł Pasteczka. 2024-02-06. Multivariable generalizations of bivariate means via invariance. https://arxiv.org/abs/2402.04121

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