Harmonic Analysis of Fractal Measures
Palle E. T. Jorgensen, Steen Pedersen
Abstract
This paper introduces Fourier duality for a class of affine iterated function systems (IFS) Ti. These systems are determined by a finite family of contractive affine maps in Rd. Our Fourier duality applies to the resulting probability measure mu which is fixed by (Ti). When the IFS is given, the support of the associated mu is a compact set X in Rd, typically a fractal. Our Fourier duality refers to the Hilbert space L2(X, mu): We show that under a certain unitarity condition involving a pair of affine iterated function systems (Ti) and (Sj) it is possible to recursively construct a Fourier bases in the Hilbert space L2(X, mu) with the Fourier basis for one depending on the other.
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