On exponential bases and frames with non-linear phase functions and some applications

TL;DR

This paper characterizes when exponential systems with non-linear phase functions form orthogonal bases or frames in various measures, revealing surprising flexibility and applications to group representations.

Contribution

It provides a complete characterization of spectral and frame-spectral exponential systems with non-linear phases, including new examples and applications.

Findings

Certain Cantor measures and the unit disk admit orthogonal bases with non-linear phases.

Only standard phases produce orthonormal bases under regularity conditions for Lebesgue measure.

A broad class of non-linear phase functions yield orthonormal bases in higher dimensions.

Abstract

In this paper, we study the spectrality and frame-spectrality of exponential systems of the type $E(\Lambda,\varphi) = \{e^{2\pi i \lambda\cdot\varphi(x)}: \lambda\in\Lambda\}$ where the phase function $\varphi$ is a Borel measurable which is not necessarily linear. A complete characterization of pairs $(\Lambda,\varphi)$ for which $E(\Lambda,\varphi)$ is an orthogonal basis or a frame for $L^{2}(\mu)$ is obtained. In particular, we show that the middle-third Cantor measures and the unit disc, each admits an orthogonal basis with a certain non-linear phase. Under a natural regularity condition on the phase functions, when $\mu$ is the Lebesgue measure on $[0,1]$ and $\Lambda= {\mathbb{Z}},$ we show that only the standard phase functions $\varphi(x) = \pm x$ are the only possible functions that give rise to orthonormal bases. Surprisingly, however we prove that there exist a greater degree of flexibility, even for continuously differentiable phase functions in higher dimensions. For instance, we were able to describe a large class of functions $\varphi$ defined on ${\mathbb{R}}^{d}$ such that the system $E(\Lambda,\varphi)$ is an orthonormal basis for $L^{2}[0,1]^{d}$ when $d\geq2.$ Moreover, we discuss how our results apply to the discretization problem of unitary representations of locally compact groups for the construction of orthonormal bases. Finally, we conclude the paper by stating several open problems.