Harmonic Analysis of Fractal Measures

الأصل · EN

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.