Quasiperiodic spectra and orthogonality for iterated function system measures

作     者：Dutkay, Dorin Ervin Jorgensen, Palle E. T. 

作者机构：Univ Cent Florida Dept Math Orlando FL 32816 USA Univ Iowa Dept Math Iowa City IA 52242 USA 

出 版 物：《MATHEMATISCHE ZEITSCHRIFT》 (数学杂志)

年 卷 期：2009年第261卷第2期

页      面：373-397页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：National Science Foundation [DMS-0704191] 

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

摘      要：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.