Related papers: Fourier bases and Fourier frames on self-affine me…
Fourier Series is the second of monographs we present on harmonic analysis. Harmonic analysis is one of the most fascinating areas of research in mathematics. Its centrality in the development of many areas of mathematics such as partial…
We consider measures supported on sets of irrational numbers possessing many consecutive partial quotients satisfying a condition based on the previous partial quotients. We show that under mild assumptions, such sets will always support…
Interferometers play an increasingly important role for spatially resolved observations. If employed at full potential, interferometry can probe an enormous dynamic range in spatial scale. Interpretation of the observed visibilities…
We construct an explicit orthonormal basis of piecewise ${}_{i+1}F_{i}$ hypergeometric polynomials for the Alpert multiresolution analysis. The Fourier transform of each basis function is written in terms of ${}_2F_3$ hypergeometric…
In this paper, we consider the stationary measure of the Hadamard walk on the one-dimensional integer lattice. Here all the stationary measures given by solving the eigenvalue problem are completely determined via the transfer matrix…
Searches for spacetime variations of fundamental constants have entered an era of unprecedented precision. New, high quality quasar spectra require increasingly refined analytic methods. In this article, a continuation in a series to…
Mathematical diffraction theory is concerned with the analysis of the diffraction measure of a translation bounded complex measure $\omega$. It emerges as the Fourier transform of the autocorrelation measure of $\omega$. The mathematically…
Fourier microscopy is becoming an increasingly important tool for the analysis of optical nanostructures and quantum emitters. However, achieving quantitative Fourier space measurements requires a thorough understanding of the impact of…
Resolving sources beyond the diffraction limit is important in imaging, communications, and metrology. Current image-based methods of super-resolution require phase information (either of the source points or an added filter) and perfect…
The framed standard model (FSM) suggested earlier, which incorporates the Higgs field and 3 fermion generations as part of the framed gauge theory structure, is here developed further to show that it gives both quarks and leptons…
We introduce probabilistic frames to study finite frames whose elements are chosen at random. While finite tight frames generalize orthonormal bases by allowing redundancy, independent, uniformly distributed points on the sphere…
We quantify the pointwise doubling properties of self-similar measures using the notion of pointwise Assouad dimension. We show that all self-similar measures satisfying the open set condition are pointwise doubling in a set of full…
The local dimension spectrum provides a framework for quantifying the fractal properties of a measure, and it is well understood for non-overlapping self-similar measures. In this article, we study the local dimension spectrum for dominated…
In the physics literature the spectral form factor (SFF), the squared Fourier transform of the empirical eigenvalue density, is the most common tool to test universality for disordered quantum systems, yet previous mathematical results have…
Cosmological random fields are often analysed in spherical Fourier-Bessel basis. Compared to the Cartesian Fourier basis this has an advantage of properly taking into account some of the relevant physical processes (redshift-space…
Mathematical diffraction theory has been developed since about 1995. Hof's initial approach relied on tempered distributions in euclidean space. Nowadays often the Fourier theory by Argabright and Gil de Lamadrid is used, which applies to…
Image analysis methods that are based on exact blur values are faced with the computational complexities due to blur measurement error. This atmosphere encourages scholars to look for handcrafted and learned features for finding depth from…
We describe a class of measurable subsets $\Omega$ in $\br^d$ such that $L^2(\Omega)$ has an orthogonal basis of frequencies $e_\lambda(x)=e^{i2\pi\lambda\cdot x}(x\in\Omega)$ indexed by $\lambda\in\Lambda\subset\br^d$. We show that such…
Equilibrium measures are special invariant measures of chaotic dynamical systems and iterated function systems, commonly studied as salient examples of fractal measures. While useful analytic expressions are rare, computational exploration…
The precision of the neutrino oscillation parameters measurements has improved and will continue to improve as the next-generation experiments become online. Beyond the more precise measurements of the mixing angles and phases used to…