Extraction and splitting of Riesz bases of exponentials

Analysis Seminar
Wednesday, March 15, 2023 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 005
David Walnut – George Mason University – dwalnut@gmu.eduhttps://science.gmu.edu/directory/david-walnut
Christopher Heil

Given a discrete set $\Lambda\subseteq\mathbb{R}$ and an interval $I$, define the sequence of complex exponentials in $L^2(I)$, $\mathcal{E}(\Lambda)$, by $\{e^{2\pi i\lambda t}\colon \lambda\in\Lambda\}$.  A fundamental result in harmonic analysis says that if $\mathcal{E}(\frac{1}{b}\mathbb{Z})$ is an orthogonal basis for $L^2(I)$ for any interval $I$ of length $b$.  It is also well-known that there exist sets $\Lambda$, which may be irregular, such that sets $\mathcal{E}(\Lambda)$ form nonorthogonal bases (known as Riesz bases) for $L^2(S)$, for $S\subseteq\mathbb{R}$ not necessarily an interval.

Given $\mathcal{E}(\Lambda)$ that forms a Riesz basis for $L^2[0,1]$ and some 0 < a < 1, Avdonin showed that there exists $\Lambda'\subseteq \Lambda$ such that $\mathcal{E}(\Lambda')$ is a Riesz basis for $L^2[0,a]$ (called basis extraction).  Lyubarskii and Seip showed that this can be done in such a way that $\mathcal{E}(\Lambda \setminus \Lambda')$ is also a Riesz basis for $L^2[a,1]$ (called basis splitting).  The celebrated result of Kozma and Nitzan shows that one can extract a Riesz basis for $L^2(S)$ from $\mathcal{E}(\mathbb{Z})$ where $S$ is a union of disjoint subintervals of $[0,1]$.

In this talk we construct sets $\Lambda_I\subseteq\mathbb{Z}$ such that the $\mathcal{E}(\Lambda_I)$ form Riesz bases for $L^2(I)$ for corresponding intervals $I$, with the added compatibility property that unions of the sets $\Lambda_I$ generate Riesz bases for unions of the corresponding intervals.  The proof of our result uses an interesting assortment of tools from analysis, probability, and number theory.  We will give details of the proof in the talk, together with examples and a discussion of recent developments.  The work discussed is joint with Shauna Revay (GMU and Accenture Federal Services (AFS)), and Goetz Pfander (Catholic University of Eichstaett-Ingolstadt).