Fourier frequencies in affine iterated function systems |
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Authors: | Dorin Ervin Dutkay Palle E.T. Jorgensen |
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Affiliation: | a Department of Mathematics, University of Central Florida, 4000 Central Florida Blvd., PO box 161364, Orlando, FL 32816-1364, USA b Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419, USA |
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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 Rd, and the “IFS” refers to such a finite system of transformations, or functions. The iteration limits are pairs (X,μ) where X is a compact subset of Rd (the support of μ), and the measure μ 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 L2(X,μ); and (2) explicit constructions of Fourier bases from the given data defining the IFS. |
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Keywords: | Fourier series Affine fractal Spectrum Spectral measure Hilbert space Attractor |
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