Fourier series on fractals: a parallel with wavelet theory

TL;DR

This paper explores the spectral properties of Fourier series on fractal measures, drawing parallels with wavelet theory, and investigates their implications in operator theory and fractal geometry.

Contribution

It establishes a connection between Fourier analysis on fractals and wavelet analysis, extending spectral theory to measures from iterated function systems.

Findings

Orthogonality relations for Fourier frequencies on fractal measures

Parallel between fractal measure analysis and wavelet geometry

Historical insights into operator theory applications

Abstract

We study orthogonality relations for Fourier frequencies and complex exponentials in Hilbert spaces $L^2(\mu)$ with measures $\mu$ arising from iterated function systems (IFS). This includes equilibrium measures in complex dynamics. Motivated by applications, we draw parallels between analysis of fractal measures on the one hand, and the geometry of wavelets on the other. We are motivated by spectral theory for commuting partial differential operators and related duality notions. While stated initially for bounded and open regions in $\br^d$, they have since found reformulations in the theory of fractals and wavelets. We include a historical sketch with questions from early operator theory.