Spectral properties of some unions of linear spaces

TL;DR

This paper investigates the spectral properties of additive spaces formed by unions of intervals or measures on the real line, focusing on the existence and structure of exponential bases, Riesz bases, and orthonormal bases depending on measure positioning.

Contribution

It provides new insights into how measure placement affects the existence of exponential bases and characterizes bases for specific overlapping and non-overlapping additive spaces.

Findings

Non-overlapping additive spaces possess Riesz bases.

Necessary conditions for overlapping spaces to have bases.

Certain overlapping Lebesgue-type spaces have exponential orthonormal bases.

Abstract

We consider \textit{additive spaces}, consisting of two intervals of unit length or two general probability measures on ${\mathbb R}^1$, positioned on the axes in ${\mathbb R}^2$, with a natural additive measure $\rho$. We study the relationship between the exponential frames, Riesz bases, and orthonormal bases of $L^2(\rho)$ and those of its component spaces. We find that the existence of exponential bases depends strongly on how we position our measures on ${\mathbb R}^1$. We show that non-overlapping additive spaces possess Riesz bases, and we give a necessary condition for overlapping spaces. We also show that some overlapping additive spaces of Lebesgue type have exponential orthonormal bases, while some do not. A particular example is the "L" shape at the origin, which has a unique orthonormal basis up to translations of the form \[ \left\{e^{2 \pi i (\lambda_1 x_1 + \lambda_2 x_2)} : (\lambda_1, \lambda_2) \in \Lambda \right\}, \] where \[ \Lambda = \{ (n/2, -n/2) \mid n \in {\mathbb Z} \}. \]