It is known that if ( ) ∈ is a sequence of orthogonal polynomials in ([-1,1],()), then the roots are distributed according to an arcsine distribution (1 - ) for a wide variety of weights (). We connect this to a result of the Hilbert transform due to Tricomi: if ()(1 - ) ∈ (-1,1) and its Hilbert transform vanishes on (-1,1), then the function is a multiple of the arcsine distribution We also prove a localized Parseval-type identity that seems to be new: if ()(1- ) ∈ L(-1, 1) and has mean value 0 on (-1, 1), then .

In a recent paper by T. Strohmer and R. Vershynin ["A Randomized Kaczmarz Algorithm with Exponential Convergence", Journal of Fourier Analysis and Applications, published online on April 25, 2008] a "randomized Kaczmarz algorithm" is proposed for solving systems of linear equations [Formula: see text] . In that algorithm the next equation to be used in an iterative Kaczmarz process is selected with a probability proportional to ‖a(i)‖ (2). The paper illustrates the superiority of this selection method for the reconstruction of a bandlimited function from its nonuniformly spaced sampling values.In this note we point out that the reported success of the algorithm of Strohmer and Vershynin in their numerical simulation depends on the specific choices that are made in translating the underlying problem, whose geometrical nature is "find a common point of a set of hyperplanes", into a system of algebraic equations. If this translation is carefully done, as in the numerical simulation provided by Strohmer and Vershynin for the reconstruction of a bandlimited function from its nonuniformly spaced sampling values, then indeed good performance may result. However, there will always be legitimate algebraic representations of the underlying problem (so that the set of solutions of the system of algebraic equations is exactly the set of points in the intersection of the hyperplanes), for which the selection method of Strohmer and Vershynin will perform in an inferior manner.

We investigate sharp frame bounds of Gabor frames with chirped Gaussians and rectangular lattices or, equivalently, the case of the standard Gaussian and general lattices. We prove that for even redundancy and standard Gaussian window the hexagonal lattice minimizes the upper frame bound using a result by Montgomery on minimal theta functions.

We propose a refinement of the Robertson-Schrödinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition, it makes a direct connection with modern harmonic analysis, since it involves the Wigner transform and its symplectic Fourier transform, which is the radar ambiguity function. As a by-product of the refined RSUP, we derive inequalities involving the entropy and the covariance matrix of Wigner distributions. These inequalities refine the Shanon and the Hirschman inequalities for the Wigner distribution of a mixed quantum state . We prove sharp estimates which critically depend on the purity of and which are saturated in the Gaussian case.

We study the problem of stable reconstruction of the short-time Fourier transform from samples taken from trajectories in . We first investigate the interplay between relative density of the trajectory and the reconstruction property. Later, we consider spiraling curves, a special class of trajectories, and connect sampling and uniqueness properties of these sets. Moreover, we show that for window functions given by a linear combination of Hermite functions, it is indeed possible to stably reconstruct from samples on some particular natural choices of spiraling curves.