Bases of complex exponentials with restricted supports

TL;DR

This paper investigates conditions under which complex exponentials with restricted supports still form a basis for square integrable functions, showing that certain partitions and support conditions preserve the basis property.

Contribution

It demonstrates that for finite unions of intervals covering [0,1], a partition of integers exists to form a Riesz basis with restricted supports.

Findings

Existence of Riesz bases with restricted supports for unions of intervals

Partition of integer frequencies aligns with support sets to maintain basis properties

Supports can be overlapping and still preserve the basis structure

Abstract

The complex exponentials with integer frequencies form a basis for the space of square integrable functions on the unit interval. We analyze whether the basis property is maintained if the support of the complex exponentials is restricted to possibly overlapping subsets of the unit interval. We show, for example, that if $S_1, \ldots, S_K \subset [0,1]$ are finite unions of intervals with rational endpoints that cover the unit interval, then there exists a partition of $\mathbb{Z}$ into sets $\Lambda_1, \ldots, \Lambda_K$ such that $\bigcup_{k=1}^K \{ e^{2\pi i \lambda (\cdot)} \chi_{S_k} : \lambda \in \Lambda_k \}$ is a Riesz basis for $L^2[0,1]$. Here, $\chi_S$ denotes the characteristic function of $S$.