A note on exponential Riesz bases

Abstract

We prove that if $I_\ell = [a_\ell,b_\ell)$, $\ell=1, \ldots, L$, are disjoint intervals in $[0,1)$ with the property that the numbers $1, a_1, \ldots, a_L, b_1, \ldots, b_L$ are linearly independent over $\mathbb{Q}$, then there exist pairwise disjoint sets $\Lambda_\ell \subset \mathbb{Z}$, $\ell=1, \ldots, L$, such that for every $J \subset \{ 1, \ldots , L \}$, the system $\{e^{2\pi i \lambda x} : \lambda\in \cup_{\ell \in J} \, \Lambda_\ell \}$ is a Riesz basis for $L^2 ( \cup_{\ell \in J} \, I_\ell)$. Also, we show that for any disjoint intervals $I_\ell$, $\ell=1, \ldots, L$, contained in $[1,N)$ with $N \in \mathbb{N}$, the orthonormal basis $\{e^{2\pi i n x} : n \in \mathbb{Z} \}$ of $L^2[0,1)$ can be complemented by a Riesz basis $\{e^{2\pi i \lambda x} : \lambda\in\Lambda\}$ for $L^2(\cup_{\ell=1}^L \, I_{\ell})$ with some set $\Lambda \subset (\frac{1}{N} \mathbb{Z}) \backslash \mathbb{Z}$, in the sense that their union $\{e^{2\pi i \lambda x} : \lambda\in \mathbb{Z} \cup \Lambda\}$ is a Riesz basis for $L^2 ( [0,1) \cup I_1 \cup \cdots \cup I_L )$.