Exponential bases for partitions of intervals

• Published in: arXiv.org 2021-09
• Main Authors: Pfander, Goetz, Revay, Shauna, Walnut, David
• Format: Article
• Language: English
• Subjects: Exponential functions; Intervals
• ISSN: 2331-8422

For a partition of \([0,1]\) into intervals \(I_1,\ldots,I_n\) we prove the existence of a partition of \(\mathbb{Z}\) into \(\Lambda_1,\ldots, \Lambda_n\) such that the complex exponential functions with frequencies in \( \Lambda_k\) form a Riesz basis for \(L^2(I_k)\), and furthermore, that for any \(J\subseteq\{1,\,2,\,\dots,\,n\}\), the exponential functions with frequencies in \( \bigcup_{j\in J}\Lambda_j\) form a Riesz basis for \(L^2(I)\) for any interval \(I\) with length \(|I|=\sum_{j\in J}|I_j|\). The construction extends to infinite partitions of \([0,1]\), but with size limitations on the subsets \(J\subseteq \mathbb{Z}\); it combines the ergodic properties of subsequences of \(\mathbb{Z}\) known as Beatty-Fraenkel sequences with a theorem of Avdonin on exponential Riesz bases.