Fourier frequencies in affine iterated function systems

By Dorin Ervin Dutkay, Palle E. T. Jorgensen. 2006

160 citations recorded by the source corpus.

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 R d , 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 R d (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 L 2 ( X , μ ) ; and (2) explicit constructions of Fourier bases from the given data defining the IFS.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1244ffd0850e8cc718239fc21f3231f7489ea39d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Spectra of a class of self-affine measures (2011, background): “1 holds in the dimension n = 1 [14], in the case when |D| = |S| = |det(M)| [17] and in the case when |det(M)| is a prime number [16], it also holds in higher dimensions with additional conditions [1,4].”
FOURIER SERIES ON FRACTALS: A PARALLEL WITH WAVELET THEORY (2007, background): “One of the conclusions of our work ([DJ06a, DJ07b , JP94, JP98]) is to establish that despite the chaos features of the attractors X, there is nonetheless a substitute for Haar measure on these sets X. Now the measures from Theorem 2.8 form a simplex of measures µp indexed by p, where p = (pi) N=1 ”