Probability and Fourier Duality for Affine Iterated Function Systems
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Probability and Fourier Duality for Affine Iterated Function Systems Dorin Ervin Dutkay · Palle E.T. Jorgensen
Received: 21 August 2008 / Accepted: 17 November 2008 / Published online: 2 December 2008 © Springer Science+Business Media B.V. 2008
Abstract Let d be a positive integer, and let μ be a finite measure on Rd . In this paper we ask when it is possible to find a subset in Rd such that the corresponding complex exponential functions eλ indexed by are orthogonal and total in L2 (μ). If this happens, we say that (μ, ) is a spectral pair. This is a Fourier duality, and the x-variable for the L2 (μ)-functions is one side in the duality, while the points in is the other. Stated this way, the framework is too wide, and we shall restrict attention to measures μ which come with an intrinsic scaling symmetry built in and specified by a finite and prescribed system of contractive affine mappings in Rd ; an affine iterated function system (IFS). This setting allows us to generate candidates for spectral pairs in such a way that the sets on both sides of the Fourier duality are generated by suitably chosen affine IFSs. For a given affine setup, we spell out the appropriate duality conditions that the two dual IFS-systems must have. Our condition is stated in terms of certain complex Hadamard matrices. Our main results give two ways of building higher dimensional spectral pairs from combinatorial algebra and spectral theory applied to lower dimensional systems. Keywords Iterated function system · Fourier · Fourier decomposition · Hilbert space · Orthogonal basis · Spectral duality · Dynamical system · Path-space measure · Spectrum · Infinite product Mathematics Subject Classification (2000) 28C15 · 30C40 · 37A60 · 42B35 · 42C05 · 46A32 · 47L50
Research supported in part by a grant from the National Science Foundation DMS-0704191. D.E. Dutkay () Department of Mathematics, University of Central Florida, 4000 Central Florida Blvd., P.O. Box 161364, Orlando, FL 32816-1364, USA e-mail: [email protected] P.E.T. Jorgensen Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419, USA e-mail: [email protected]
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D.E. Dutkay, P.E.T. Jorgensen
1 Introduction The use of traditional Fourier series has up to recently been restricted to the setting of Fourier duality between groups; in the Abelian case [25], between compact groups (such as tori) on the one side, each group coming with its Haar measure; and discrete Abelian groups (such as lattices) on the other. However in dynamics and in other applications to computational mathematics, one is often faced with sets arising as attractors, highly non-linear, and coming equipped with equilibrium measures. This has led to attempts at adapting traditional Fourier tools to these non-linear and non-group settings. In this paper we address the Fourier duality question for affine iterated function systems. For some of the earlier literature we refer the reader to [2, 5–9, 11–17, 19, 24]. Iterated function systems (IFS) in Rd are natural gener
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