Dominating sets of frequencies in spectrums of measures with finite energy

S. V. Khrushchev

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part X, Tome 107 (1982), pp. 222-227

Résumé

A subset $\Lambda$ of $\mathbb Z$ is called a dominating set if every measure $\mu$, satisfying $\sum_{n\in\Lambda\setminus\{0\}}(|\hat\mu(n)|^2)(|n|)+\infty$, has a finite energy $\varepsilon(\mu)=\sum_{n\in\mathbb Z\setminus\{0\}}(|\hat\mu(n)|^2)(|n|)+\infty$. It is proved that a low density of a dominating set is positive and for every $\varepsilon>0$ there is a dominating set $\Lambda$, $\Lambda\subset\mathbb Z_+$, whose density is smaller than $\varepsilon$.