Fourier series on fractals: a parallel with wavelet theory

Details Fourier series on fractals: a parallel with wavelet theory Fourier series on fractals: a parallel with wavelet theory

2007-09-17

arxiv.org/abs/0709.2702v2

Mathematics

Functional Analysis

v2, minor correction in section 4

Scientific paper

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.

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