A note on exponential Riesz bases

TL;DR

This paper demonstrates the construction of Riesz bases of exponential functions for unions of disjoint intervals within [0,1), under certain linear independence conditions, extending the classical Fourier basis to more complex domains.

Contribution

It establishes the existence of Riesz bases of exponential functions for unions of disjoint intervals, generalizing classical Fourier bases and providing new tools for harmonic analysis on irregular domains.

Findings

Existence of Riesz bases for unions of disjoint intervals under linear independence conditions.

Construction of complementary Riesz bases in [0,1) for unions of intervals.

Union of these bases forms a Riesz basis for the combined domain.

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 $L^2 ( [0,1) \cup I_1 \cup \cdots \cup I_L )$.