Title

On The Spectra Of A Cantor Measure

Keywords

Affine fractals; Attractor; Fourier series; Hilbert spaces; Spectral measure; Spectrum

Abstract

We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75 (1998) 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.

Publication Date

5-1-2009

Publication Title

Advances in Mathematics

Volume

221

Issue

1

Number of Pages

251-276

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.aim.2008.12.007

Socpus ID

62249112799 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/62249112799

This document is currently not available here.

Share

COinS