Title

Quasiperiodic Spectra And Orthogonality For Iterated Function System Measures

Keywords

Aperiodic tilings; Fourier series; Fractals; Hadamard matrix; Hilbert space; Iterated function systems; Matrix algorithms; Nonlinear analysis; Orthogonal bases; Quasicrystals; Riesz products; Special orthogonal functions

Abstract

We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension. Moreover, the spectrum is then typically quasi-periodic, but non-periodic, i.e., the spectrum is a "small perturbation" of a lattice. Due to earlier research on IFSs, there are known results on certain classes of spectral duality-pairs, also called spectral pairs or spectral measures. It is known that some duality pairs are associated with complex Hadamard matrices. However, not all IFSs X admit spectral duality. When X is given, we identify geometric conditions on X for the existence of a Fourier spectrum, serving as the second part in a spectral pair. We show how these spectral pairs compose, and we characterize the decompositions in terms of atoms. The decompositions refer to tensor product factorizations for associated complex Hadamard matrices. © 2008 Springer-Verlag.

Publication Date

2-1-2009

Publication Title

Mathematische Zeitschrift

Volume

261

Issue

2

Number of Pages

373-397

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s00209-008-0329-2

Socpus ID

56349147498 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/56349147498

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