A class of sequences with a strong average property
A sequence \((m_ k)\) of probability measures on \({\mathbb{N}}\) is called ergodic if lim \(<f,m_ k>=<f,\lambda >\) holds for each continuous f on the Bohr compactification b\({\mathbb{Z}}\) of \({\mathbb{Z}}\). A bounded sequence (g(n)) is called p-sequence, if lim \(<g,m_ k>=p\) for each ergodic sequence \((m_ k)\) on \({\mathbb{N}}\). It is shown that (g(n)) is a p-sequence for some \(p\in {\mathbb{R}}\) if and only if there exist, for each \(\epsilon >0\), measures \(\mu_ 1\), \(\mu_ 2\) on the torus \(T={\mathbb{R}}/{\mathbb{Z}}\) with \[ {\hat \mu}_ 1\leq g\leq {\hat \mu}_ 2\quad on\quad {\mathbb{N}}\quad and\quad \lim N^{-1}\sum^{N}_{1}({\hat \mu}_ 2(n)-{\hat \mu}_ 1(n))<\epsilon. \] (\({\hat \mu}\) denotes the Fourier transform).
scientific article; zbMATH DE number 3938131
scientific article