  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 $f(x)=\sum_{n=0}^{\infty}c_ne^{2\pi inx}$. 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.
