Some properties of spectral measures

Łaba, Izabella; Yang, Wang

doi:10.1016/j.acha.2005.03.003

Cited by 56 publications

(29 citation statements)

References 17 publications

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“…In contrast, the role of the examples in spectral theory and in Fourier duality is of a later vintage (see e.g., [22,28,30,38]) and there the stability properties are quite different as can be seen from Sect. 5.…”

Section: Introductionmentioning

confidence: 89%

Analysis of orthogonality and of orbits in affine iterated function systems

2007

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We introduce a duality for affine iterated function systems (AIFS) which is naturally motivated by group duality in the context of traditional harmonic analysis. Our affine systems yield fractals defined by iteration of contractive affine mappings. We build a duality for such systems by scaling in two directions: fractals in the small by contractive iterations, and fractals in the large by recursion involving iteration of an expansive matrix. By a fractal in the small we mean a compact attractor X supporting Hutchinson's canonical measure µ, and we ask when µ is a spectral measure, i.e., when the Hilbert space L 2 (µ) has an orthonormal basis (ONB) of exponentials {e λ | λ ∈ Λ}. We further introduce a Fourier duality using a matched pair of such affine systems. Using next certain extreme cycles, and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary Fourier series and which serve as spectra for X. Our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum for X. Our results in turn are illustrated with concrete Sierpinski like fractals in dimensions 2 and 3.

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Section: Introductionmentioning

confidence: 89%

Analysis of orthogonality and of orbits in affine iterated function systems

2007

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“…This general framework includes subjects such as universal tiling sets and the dual spectral set conjecture (see [11,22,25,27,35]). Starting with [16], questions about Fourier duality have received considerable attention with respect to pure harmonic analysis [5,8,9,24,29,30,33,34] and with respect to applications such as wavelets, sampling, algorithms, martingales, and substitution-dynamical systems [2][3][4]26].…”

Section: Overview Of Prior Literaturementioning

confidence: 99%

Families of Spectral Sets for Bernoulli Convolutions

Jørgensen

Kornelson

Shuman

2010

J Fourier Anal Appl

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We study the harmonic analysis of Bernoulli measures μ λ , a one-parameter family of compactly supported Borel probability measures on the real line. The parameter λ is a fixed number in the open interval (0, 1). The measures μ λ may be understood in any one of the following three equivalent ways: as infinite convolution measures of a two-point probability distribution; as the distribution of a random power series; or as an iterated function system (IFS) equilibrium measure determined by the two transformations λ(x ± 1). For a given λ, we consider the harmonic analysis in the sense of Fourier series in the Hilbert space L 2 (μ λ ). For L 2 (μ λ ) to have infinite families of orthogonal complex exponential functions e 2πis(·) , it is known that λ must be a rational number of the form m 2n , where m is odd. We show that L 2 (μ 1 2n ) has a variety of Fourier bases; i.e. orthonormal bases of exponential functions. For some other rational values of λ, we exhibit maximal Fourier families that are not orthonormal bases.

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“…Let S ⊆ Z n be a finite subset of the cardinality |S| = |D|, corresponding to the dual IFS {ψ s (x) = R * x + s} s∈S , we use (R * , S) to denote the expansive orbit of 0 under {ψ s (x)} s∈S , that is where R * is the conjugate transpose matrix of R. Recall that for a probability measure μ of compact support on R n , we call μ a spectral measure if there exists a discrete set ⊆ R n such that E := { e 2πi λ, x : λ ∈ } forms an orthogonal basis for L 2 (μ). The set is then called a spectrum for μ. Spectral measure is a natural generalization of spectral set introduced by Fuglede [4] whose famous conjecture and its related problems have received much attention in recent years(see [17,21]). The spectral self-affine measure problem at the present day consists in determining conditions under which μ M,D is a spectral measure, and has been studied in the papers [1,2,13,16,[19][20][21][22] (see also [23,24] for the main goal).…”

Section: Introductionmentioning

confidence: 99%