A note on exponential Riesz bases

Abstract

We prove that if $I_{\ell }=[a_{\ell },b_{\ell })$, $\ell =1,\dots ,L$, are disjoint intervals in $[0,1)$ with the property that the numbers $1,a_1,\dots ,a_L,b_1,\dots ,b_L$ are linearly independent over $\mathbb{Q}$, then there exist pairwise disjoint sets ${\Lambda }_{\ell }\subset \mathbb{Z}$, $\ell =1,\dots ,L$, such that for every $J\subset \{1,\dots ,L\}$, 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 $I_{\ell }$, $\ell =1,\dots ,L$, contained in $[1,N)$ with $N\in \mathbb{N}$, the orthonormal basis $\{e^{2\pi inx}:n\in \mathbb{Z}\}$ 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 $L^2([0,1)\cup I_1\cup \cdots \cup I_L)$.

References in corpus (4)

• Combining Riesz bases in $R^d$
• Exponential bases for partitions of intervals
• A set with no Riesz basis of exponentials
• On construction of bounded sets not admitting a general type of Riesz spectrum

Cited by in corpus (1)

• Kadec-type theorems for sampled group orbits