On the spectra of a Cantor measure

TL;DR

This paper characterizes all orthonormal exponential bases on the Cantor measure, introduces a spectral labeling framework, and provides new conditions for spectrum generation, expanding the known set of orthogonal bases and revealing new phenomena.

Contribution

It offers a complete characterization of orthonormal exponential bases on the Cantor measure via spectral labelings and introduces new conditions for spectrum construction, extending previous results.

Findings

Established a one-to-one correspondence between spectral labelings and orthonormal bases.

Provided a sufficient condition for spectral labelings to generate a spectrum.

Discovered examples where maximal orthogonal sets are not bases.

Abstract

We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75,1998, pp 185-228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one-to-one correspondence with the spectral labelings of the infinite binary tree. With the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum (an orthonormal basis). This result not only provides us an easy and efficient way to construct various of new spectra for the Cantor measure but also extends many previous results in the literature. In fact, most known examples of orthonormal bases of exponentials correspond to spectral labelings satisfying this sufficient condition. We also obtain two new conditions for a labeling tree to generate a spectrum when other digits (digits not necessarily in $\{0, 1, 2, 3\}$) are used in the base 4 expansion of integers and when bad branches are allowed in the spectral labeling. These new conditions yield new examples of spectra and in particular lead to a surprizing example which shows that a maximal set of orthogonal exponentials is not necessarily an orthonormal basis.