Exponential bases for partitions of intervals

Abstract

For a partition of $[0,1]$ into intervals $I_1,\dots ,I_n$ we prove the existence of a partition of $Z$ into ${\Lambda }_1,\dots ,{\Lambda }_n$ such that the complex exponential functions with frequencies in ${\Lambda }_k$ form a Riesz basis for $L^2\left(I_k\right)$, and furthermore, that for any $J\subseteq \{1,2,\dots ,n\}$, the exponential functions with frequencies in ${\bigcup }_{j\in J}{\Lambda }_j$ form a Riesz basis for $L^2\left(I\right)$ for any interval I with length $\left|I\right|={\sum }_{j\in J}\left|I_j\right|$. The construction extends to infinite partitions of $[0,1]$, but with size limitations on the subsets $J\subseteq Z$; it combines the ergodic properties of subsequences of $Z$ known as Beatty-Fraenkel sequences with a theorem of Avdonin on exponential Riesz bases.