Title

On the spectra of a Cantor measure

Authors

Authors

D. E. Dutkay; D. G. Han;Q. Y. Sun

Comments

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Abbreviated Journal Title

Adv. Math.

Keywords

Fourier series; Affine fractals; Spectrum; Spectral measure; Hilbert; spaces; Attractor; ITERATED FUNCTION SYSTEMS; MOCK FOURIER-SERIES; FUGLEDES CONJECTURE; SET; CONJECTURE; OPERATORS; Mathematics

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. Published by Elsevier Inc.

Journal Title

Advances in Mathematics

Volume

221

Issue/Number

1

Publication Date

1-1-2009

Document Type

Article

Language

English

First Page

251

Last Page

276

WOS Identifier

WOS:000265249000007

ISSN

0001-8708

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