We study divergence properties of the Fourier series on Cantor-type fractal measures, also called the mock Fourier series. We show that in some cases the $L^1$ -norm of the corresponding Dirichlet kernel grows exponentially fast, and therefore the Fourier series are not even pointwise convergent. We apply these results to the Lebesgue measure to show that a certain rearrangement of the exponential functions, with affine structure, which we call a scrambled Fourier series, have a corresponding Dirichlet kernel whose $L^1$ -norm grows exponentially fast, which is much worse than the known logarithmic bound. The divergence properties are related to the Mahler measure of certain polynomials and to spectral properties of Ruelle operators.

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.

The theory of Gabor frames for the infinite dimensional signal/ function space <!-- MATH $L^{2}(\mathbb{R}^d)$ --> and for the finite dimensional signal space <!-- MATH $\mathbb{R}^{d}$ --> (or <!-- MATH $\mathbb{C}^{d}$ --> ) has been extensively investigated in the last twenty years. However, very little has been done for the Gabor theory in the infinite dimensional discrete signal space <!-- MATH $\ell ^2(\mathbb{Z}^d)$ --> , especially when 1$">. In this paper we investigate the general theory for discrete Gabor frames in <!-- MATH $\ell ^2(\mathbb{Z}^d)$ --> . We focus on a few fundamental aspects of the theory such as the density/incompleteness theorem for frames and super-frames, the characterizations for dual frame pairs and orthogonal (strongly disjoint) frames, and the existence theorem for the tight dual frame of the Gabor type. The existence result for Gabor frames (resp. super-frames) requires a generalization of a standard result on common subgroup coset representatives.