On the spectra of a Cantor measure

    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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