Title: Harmonic Analysis on Fractals
Abstract: We study spectral duality for singular measures \mu. Complex Hadamard matrices is one source of such examples, and there are others. The main question is to decide when L^2( \mu) will have an orthogonal Fourier basis; i.e., when is there a fractal Fourier transform? Not so for the middle-third Cantor! Nonetheless, Jorgensen and Pedersen [JoPe98] showed that spectral duality does hold for the middle-1/4 Cantor measure. Higher dimensional L^2 fractals are associated to certain Complex Hadamard matrices.
For affine fractals, the distribution of Fourier frequencies satisfies very definite lacunary properties, in the form of geometric almost gaps; the size of the gaps grows exponentially, with sparsity between partitions.
Motivated by wavelet analysis (on fractals), R. Strichartz showed (shortly after [JoPe98] ) that these lacunary Fourier series offer better convergence properties than the classical counterparts; one reason is that, like wavelets, they are better localized. Another family of Cantor spaces we study arise as limits of infinite discrete models; e.g., infinite weighted graphs from electrical networks with resistors. Then the Cantor spaces arise as boundaries; for example, Poisson boundaries, Shilov boundaries, Martin boundaries, path-space boundaries, and metric boundaries.