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A note on exponential Riesz bases (2110.07988v2)

Published 15 Oct 2021 in math.CA

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 $L2 ( \cup_{\ell \in J} \, I_\ell)$. 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 $L2[0,1)$ can be complemented by a Riesz basis ${e{2\pi i \lambda x} : \lambda\in\Lambda}$ for $L2(\cup_{\ell=1}L \, I_{\ell})$ 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 $L2 ( [0,1) \cup I_1 \cup \cdots \cup I_L )$.

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