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Fourier Series for Singular Measures (1503.04856v3)
Published 16 Mar 2015 in math.FA
Abstract: Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure $\mu$ on $[0,1)$, every $f\in L2(\mu)$ 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.