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.

In this paper, we consider nonsubsampled graph filter banks (NSGFBs) to process data on a sparse graph. The analysis filter banks of NSGFBs have small bandwidth, pass/block the normalized constant signal, and have stability on $\ell^2$ . Given an analysis filter bank with small bandwidth, we introduce algebraic and optimization methods to construct well-localized synthesis filter banks such that the corresponding NSGFBs provide a perfect signal reconstruction in the noiseless setting. In this paper, we also prove that the proposed NSGFBs can control the resonance effect in the presence of bounded noise and they can limit the influence of shot noise primarily to a small neighborhood near its location on the graph. We later introduce an iterative distributed algorithm to implement the proposed NSGFBs, and we develop an NSGFB-based denoising technique which is demonstrated to have satisfactory performance on noise suppression.

We consider convolution sampling and reconstruction of signals in certain reproducing kernel subspaces of $L^p, 1\le p\le \infty$ . We show that signals in those subspaces could be stably reconstructed from their convolution samples taken on a relatively separated set with small gap. Exponential convergence and error estimates are established for the iterative approximation-projection reconstruction algorithm.