Received: 21 February 2017 / Revised: 23 March 2017 / Accepted: 24 March 2017 / Published: 28 March 2017

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**Abstract**

Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure $\mu $ on $[0,1)$ , every $f\in {L}^{2}\left(\mu \right)$ possesses a Fourier series of the form $f\left(x\right)={\sum}_{n}^{}$

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Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure $\mu $ on $[0,1)$ , every $f\in {L}^{2}\left(\mu \right)$ possesses a Fourier series of the form $f\left(x\right)={\sum}_{n=0}^{\infty}{c}_{n}{e}^{2\pi inx}$ . We show that the coefficients ${c}_{n}$ can be computed in terms of the quantities $\widehat{f}\left(n\right)={\int}_{0}^{1}f\left(x\right){e}^{-2\pi inx}d\mu \left(x\right)$ . We also demonstrate a Shannon-type sampling theorem for functions that are in a sense $\mu $ -bandlimited.
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(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)