It has been known for some time that complex Hadamard matrices are important in quantum physics, where they appear as a recurrent theme in diverse contexts; especially in measurements, and in uncertainty-considerations. In fractal analysis, seemingly unrelated to quantum physics, complex Hadamard matrices appear as follows: For every complex Hadamard matrix there are associated (typically) singular probability measures \mu , with self-similarities dictated by the complex Hadamard matrix. The family of measures \mu has a dual pair of symmetries in time-frequency, also dictated by the given complex Hadamard matrix. The study of spectral duality for singular measures started with a joint paper [JoPe98] between a former student of mine S. Pedersen and myself. When the specified complex Hadamard matrix further satisfies an additional symmetry; then the L^2 space of \mu will have an orthogonal Fourier basis; in other words we get an associated fractal Fourier transform; a fact that surprised classical harmonic analysts at the time. In order to appreciate the nature of the spectral duality, [JoPe98] showed that spectral duality holds for the middle-1/4 Cantor measure, but not for the middle-1/3 cousin. Typically the distribution of the associated Fourier frequencies satisfies very definite lacunar properties, in the form of geometric almost gaps; the size of the gaps grows exponentially, with sparsity between partitions. Folks in harmonic analysis have fun with them; for example R. Strichartz showed (shortly after [JoPe98] ) that these lacunar Fourier series has better convergence properties than the classical counterparts; one reason is that they are better localized.
Fractal Fourier analysis