Iterated quasi-arithmetic mean-type mappings

Paweł Pasteczka¹

¹ Institute of Mathematics Pedagogical University of Cracow Podchorążych 2 30-084 Kraków, Poland

Colloquium Mathematicum, Tome 144 (2016) no. 2, pp. 215-228

Résumé

We work with a fixed $N$-tuple of quasi-arithmetic means $M_1,\dots ,M_N$ generated by an $N$-tuple of continuous monotone functions $f_1,\dots ,f_N: I \to \mathbb {R}$ ($I$ an interval) satisfying certain regularity conditions. It is known [initially Gauss, later Gustin, Borwein, Toader, Lehmer, Schoenberg, Foster, Philips et al.] that the iterations of the mapping $I^N \ni b \mapsto (M_1(b),\dots ,M_N(b))$ tend pointwise to a mapping having values on the diagonal of $I^N$. Each of [all equal] coordinates of the limit is a new mean, called the Gaussian product of the means $M_1,\dots ,M_N$ taken on $b$. We effectively measure the speed of convergence to that Gaussian product by producing an effective—doubly exponential with fractional base—majorization of the error.

Détail

DOI : 10.4064/cm6479-2-2016

Keywords: work fixed n tuple quasi arithmetic means dots generated n tuple continuous monotone functions dots mathbb interval satisfying certain regularity conditions known initially gauss later gustin borwein toader lehmer schoenberg foster philips iterations mapping mapsto dots tend pointwise mapping having values diagonal each equal coordinates limit mean called gaussian product means dots taken nbsp effectively measure speed convergence gaussian product producing effective doubly exponential fractional base majorization error