Fourier Series for Singular Measures

Abstract

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 $f(x)={\sum }_{n=0}^{\infty }{c}_n{e}^{2\pi inx}$. We show that the coefficients $c_n$ can be computed in terms of the quantities $\hat{f}(n)={\int }_0^{1}f(x)e^{-2\pi inx}d\mu (x)$. We also demonstrate a Shannon-type sampling theorem for functions that are in a sense $\mu$-bandlimited.

12 pages