Fourier Series for Singular Measures

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