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