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arXiv · 1503.04856

Fourier Series for Singular Measures

John E. Herr·Eric S. Weber
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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 $f(x)=\sum_{n=0}^{\infty}c_ne^{2\pi inx}$. 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.

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John E. Herr, Eric S. Weber. 2015-03-16. Fourier Series for Singular Measures. https://arxiv.org/abs/1503.04856

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