FOURIER DUALITY FOR FRACTAL MEASURES WITH AFFINE SCALES

    Authors

    D. E. Dutkay;P. E. T. Jorgensen

    Comments

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    Abbreviated Journal Title

    Math. Comput.

    Keywords

    Affine fractal; Cantor set; Cantor measure; iterated function system; Hilbert space; Beurling density; Fourier bases; ITERATED FUNCTION SYSTEMS; COMPLEX HADAMARD-MATRICES; LARGEST PRIME; FACTOR; BEURLING DIMENSION; SPECTRAL PROBLEM; CANTOR MEASURES; CONJECTURE; OPERATORS; ORTHOGONALITY; ENDOMORPHISMS; Mathematics, Applied

    Abstract

    For a family of fractal measures, we find an explicit Fourier duality. The measures in the pair have compact support in R-d, and they both have the same matrix scaling; but the two use different translation vectors, one by a subset B in R-d. and the other by a related subset L. Among other things, we show that there is then a pair of infinite discrete sets Gamma(L) and Gamma(B) in R-d such that the Gamma(L)-Fourier exponentials are orthogonal in L-2(mu(B)), and the Gamma(B)-Fourier exponentials are orthogonal in L-2(mu(L)). These sets of orthogonal "frequencies" are typically lacunary, and they will be obtained by scaling in the large. The nature of our duality is explored below both in higher dimensions and for examples on the real line. Our duality pairs do not always yield orthonormal Fourier bases in the respective L-2(mu)-Hilbert spaces, but depending on the geometry of certain finite orbits, we show that they do in some cases. We further show that there are new and surprising scaling symmetries of relevance for the ergodic theory of these affine fractal measures.

    Journal Title

    Mathematics of Computation

    Volume

    81

    Issue/Number

    280

    Publication Date

    1-1-2012

    Document Type

    Article

    Language

    English

    First Page

    2253

    Last Page

    2273

    WOS Identifier

    WOS:000309315200015

    ISSN

    0025-5718

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