Fourier frequencies in affine iterated function systems

    Authors

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

    Comments

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    Abbreviated Journal Title

    J. Funct. Anal.

    Keywords

    fourier series; affine fractal; spectrum; spectral measure; Hilbert; space; attractor; HARMONIC-ANALYSIS; SPECTRAL THEORY; CANTOR MEASURES; SETS; FRACTALS; DIMENSIONS; OPERATORS; SERIES; TILES; Mathematics

    Abstract

    We examine two questions regarding Fourier frequencies for a class of iterated function systems (IFS). These are iteration limits arising from a fixed finite families of affine and contractive mappings in R-d, and the "IFS" refers to such a finite system of transformations, or functions. The iteration limits are pairs (X, mu) where X is a compact subset of R-d (the support of mu), and the measure mu is a probability measure determined uniquely by the initial IFS mappings, and a certain strong invariance axiom. The two questions we study are: (1) existence of an orthogonal Fourier basis in the Hilbert space L-2(X, mu); and (2) explicit constructions of Fourier bases from the given data defining the IFS. (C) 2007 Elsevier Inc. All rights reserved.

    Journal Title

    Journal of Functional Analysis

    Volume

    247

    Issue/Number

    1

    Publication Date

    1-1-2007

    Document Type

    Article

    Language

    English

    First Page

    110

    Last Page

    137

    WOS Identifier

    WOS:000246633000003

    ISSN

    0022-1236

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