Fourier Series for Singular Measures

Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure $μ$ on $[0,1)$, every $f\in L^2(μ)$ possesses a Fourier series of the form $f(x)=\sum_{n=0}^{\infty}c_ne^{2πinx}$. We show that the coefficients $c_{n}$ can be computed in terms of the quantities $\hat{f}(n) = \int_{0}^{1} f(x) e^{-2πi n x} d μ(x)$. We also demonstrate a Shannon-type sampling theorem for functions that are in a sense $μ$-bandlimited.

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