Motivated by the existence problem of Fourier frames on fractal measures, we introduce Bessel and frame measures for a given finite measure on <!-- MATH $\mathbb{R}^d$ --> , as extensions of the notions of Bessel and frame spectra that correspond to bases of exponential functions. Not every finite compactly supported Borel measure admits frame measures. We present a general way of constructing Bessel/frame measures for a given measure. The idea is that if a convolution of two measures admits a Bessel measure, then one can use the Fourier transform of one of the measures in the convolution as a weight for the Bessel measure to obtain a Bessel measure for the other measure in the convolution. The same is true for frame measures, but with certain restrictions. We investigate some general properties of frame measures and their Beurling dimensions. In particular, we show that the Beurling dimension is invariant under convolution (with a probability measure) and under a certain type of discretization. Moreover, if a measure admits a frame measure, then it admits an atomic one, and hence a weighted Fourier frame. We also construct some examples of frame measures for self-similar measures.

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.