Luís Daniel Abreu

A q-sampling theorem related to the q-Hankel transform

A q-version of the sampling theorem is derived using the q-Hankel transform introduced by Koornwinder and Swarttouw. The sampling points are the zeros of the third Jackson q-Bessel function.

THE ROOTS OF THE THIRD JACKSON q-BESSEL FUNCTION

We derive analytic bounds for the zeros of the third Jackson q -Bessel function J v ( 3 ) ( z ; q ) .

Function Spaces of Polyanalytic Functions

This article is meant as both an introduction and a review of some of the recent developments on Fock and Bergman spaces of polyanalytic functions. The study of polyanalytic functions is a classic topic in complex analysis. However, thanks to the interdisciplinary transference of knowledge promoted within the activities of HCAA network it has benefited from a cross-fertilization with ideas from signal analysis, quantum physics, and random matrices. We provide a brief introduction to those ideas and describe some of the results of the mentioned cross-fertilization. The departure point of our investigations is a thought experiment related to a classical problem of multiplexing of signals, in order words, how to send several signals simultaneously using a single channel.

Banach Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group

Gabor frames with Hermite functions are equivalent to sampling sequences in true Fock spaces of polyanalytic functions. In the L 2-case, such an equivalence follows from the unitarity of the polyanalytic Bargmann transform. We will introduce Banach spaces of polyanalytic functions and investigate the mapping properties of the polyanalytic Bargmann transform on modulation spaces. By applying the theory of coorbit spaces and localized frames to the Fock representation of the Heisenberg group, we derive explicit polyanalytic sampling theorems which can be seen as a polyanalytic version of the lattice sampling theorem discussed by J.M. Whittaker in Chapter 5 of his book Interpolatory Function Theory.

FUNCTIONS q-ORTHOGONAL WITH RESPECT TO THEIR OWN ZEROS

In 1939, G. H. Hardy proved that, under certain conditions, the only functions satisfying 1 0 f(λ m t)f(λ n t)dt= 0, where the λ n are the zeros of f, are the Bessel functions. We replace the above integral by the Jackson g-integral and give the q-analogue of Hardys result.

Discrete coherent states for higher Landau levels

We consider the quantum dynamics of a charged particle evolving under the action of a constant homogeneous magnetic field, with emphasis on the discrete subgroups of the Heisenberg group (in the Euclidean case) and of the SL(2, R) group (in the Hyperbolic case). We investigate completeness properties of discrete coherent states associated with higher order Euclidean and hyperbolic Landau levels, partially extending classic results of Perelomov and of Bargmann, Butera, Girardello and Klauder. In the Euclidean case, our results follow from identifying the completeness problem with known results from the theory of Gabor frames. The results for the hyperbolic setting follow by using a combination of methods from coherent states, time-scale analysis and the theory of Fuchsian groups and their associated automorphic forms.

An inverse problem for localization operators

A classical result of time–frequency analysis, obtained by Daubechies in 1988, states that the eigenfunctions of a time–frequency localization operator with circular localization domain and Gaussian analysis window are the Hermite functions. In this contribution, a converse of Daubechies’ theorem is proved. More precisely, it is shown that, for simply connected localization domains, if one of the eigenfunctions of a time–frequency localization operator with Gaussian window is a Hermite function, then its localization domain is a disc. The general problem of obtaining, from some knowledge of its eigenfunctions, information about the symbol of a time–frequency localization operator is denoted as the inverse problem, and the problem studied by Daubechies as the direct problem of time–frequency analysis. Here, we also solve the corresponding problem for wavelet localization, providing the inverse problem analogue of the direct problem studied by Daubechies and Paul.

The Weyl–Heisenberg ensemble: hyperuniformity and higher Landau levels

Weyl-Heisenberg ensembles are a class of determinantal point processes associated with the Schrodinger representation of the Heisenberg group. Hyperuniformity characterizes a state of matter for which (scaled) density fluctuations diminish towards zero at the largest length scales. We will prove that Weyl-Heisenberg ensembles are hyperuniform. Weyl-Heisenberg ensembles include as a special case a multi-layer extension of the Ginibre ensemble modeling the distribution of electrons in higher Landau levels, which has recently been object of study in the realm of the Ginibre-type ensembles associated with polyanalytic functions. In addition, the family of Weyl-Heisenberg ensembles includes new structurally anisotropic processes, where point-statistics depend on the different spatial directions, and thus provide a first means to study directional hyperuniformity.

Measures of localization and quantitative Nyquist densities

Abstract We obtain a refinement of the degrees of freedom estimate of Landau and Pollak. More precisely, we estimate, in terms of ϵ , the increase in the degrees of freedom resulting upon allowing the functions to contain a certain prescribed amount of energy ϵ outside a region delimited by a set T in time and a set Ω in frequency. In this situation, the lower asymptotic Nyquist density | T | | Ω | / 2 π is increased to ( 1 + ϵ ) | T | | Ω | / 2 π . At the technical level, we prove a pseudospectra version of the classical spectral dimension result of Landau and Pollak, in the multivariate setting of Landau. Analogous results are obtained for Gabor localization operators in a compact region of the time-frequency plane.

A Paley-Wiener theorem for the Askey-Wilson function transform

We define an analogue of the Paley-Wiener space in the context of the Askey-Wilson function transform, compute explicitly its reproducing kernel and prove that the growth of functions in this space of entire functions is of order two and type lnq 1 , providing a Paley-Wiener Theorem for the Askey-Wilson transform. Up to a change of scale, this growth is related to the refined concepts of exponential order and growth proposed by J. P. Ramis. The Paley-Wiener theorem is proved by combining a sampling theorem with a result on interpolation of entire functions due to M. E. H. Ismail and D. Stanton.