A dynamical characterization of quasilinear means
Let \(D\left( \left[ 0,1\right] \right) =\left\{ m/2^{n}:m=0,1,\dots,2^{n};n\in \mathbb{N}_{0}\right\} \) denote the set of dyadic numbers of the interval \( \left[ 0,1\right] .\) Let \(I\) be a real interval and \(F\) be a function defined on \(I\times I\) such that \(F\left( I\times I\right) \sqsubseteq I.\) Fix a pair \(x,y\in I.\) A family \(\left\{ F^{d}(x,y):d\in D\left( \left[ 0,1 \right] \right) \right\} \) of dyadic iterates of \(F\) on \(\left[ x,y\right] \) is defined as follows. Firstly, set \(F^{0}(x,y)\equiv x,\) \(F^{1}(x,y)\equiv y.\) Now, assume that \(F^{\frac{j}{2^{n}}}(x,y)\) is known for \(n\geq 0\) and for every \(0\leq j\leq 2^{n}.\) If \(k=2h\) with \(0\leq h\leq 2^{n},\) we have \( F^{\frac{k}{2^{n+1}}}(x,y)=F^{\frac{h}{2^{n}}}(x,y).\) If \(k=2h+1\) with \( 0\leq h\leq 2^{n}-1,\) define \(F^{\frac{k}{2^{n+1}}}(x,y)=F\left( F^{\frac{h}{ 2^{n}}}(x,y),F^{\frac{h+1}{2^{n}}}(x,y)\right).\) The author proves that if \(F\) is a continuous mean \(M,\) the dyadic iterations \(M^{d}\) are extended to a family of continuos means \(\left\{ M^{\delta },\delta \in \left[ 0,1\right] \right\},\) where \(M^{\delta }(x,y)=\lim_{d_{n}\nearrow \delta }M^{d_{n}}(x,y)\) for an increasing sequence \(\left\{ d_{n}\right\} \sqsubseteq D\left( \left[ 0,1\right] \right).\) Moreover, denoting by \(P\) the point \(\left( x,y\right) \in I^{2},\) it is shown that for each pair \( u,v\in \left[ 0,1\right] ,\) there exists a unique \(\mu _{P}(u,v)\in \left[ 0,1\right] \) such that \(M\left( M^{u}(x,y),M^{v}(x,y)\right) =M^{\mu _{P}(u,v)}(x,y).\) It is also proved that \(\left\{ \mu _{P}:P\in I^{2}\right\} \) is a family of means, called base family of the mean \(M.\) If the mean \(\mu _{P}\) does not depend on the point \(P,\) thus \(\mu _{P}=\mu \) for each point \(P\in I^{2},\) it is said that a base mean is admitted by \(M\) and that \(\mu \) is a base mean of \(M.\) The main result of the paper is that for a differentiable strict mean \(M,\) a base mean is admitted by \(M\) if and only if \(M\) is a quasilinear mean, thus it is defined by \(M(x,y)=\phi ^{-1}((1-\alpha )\phi (x)+\alpha \phi (y))\), for \(x,y\in I,\) where \(\alpha \in \left( 0,1\right) \) and \(\phi :I\rightarrow \mathbb{R}\) is a strictly monotonic and continuous function.
scientific article; zbMATH DE number 6109262