A class of scaled Bessel sampling theorems

Sampling theorems for a class of scaled Bessel unitary transforms are presented. The derivations are based on the properties of the generalized Laguerre functions. This class of scaled Bessel unitary transforms includes the classical sine and cosine transforms, but also novel chirp sine and modified Hankel transforms. The results for the sine and cosine transform can also be utilized to yield a sampling theorem, different from Shannon's, for the Fourier transform

Landau's necessary density conditions for the Hankel transform

We will prove an analogue of Landau's necessary conditions [Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967).] for spaces of functions whose Hankel transform is supported in a measurable subset S of the positive semi-axis. As a special case, necessary density conditions for the existence of Fourier-Bessel frames are obtained.Comment: To appear on J. Funct. Analysi

Amostragem uniforme e não-uniforme

Mestrado em Psicologia ForenseNeste trabalho vamos apresentar alguns conceitos básicos da teoria da amostragem uniforme e da não-uniforme, e sua ligação à teoria de interpolação. Numa primeira parte focaremos essencialmente os conceitos de amostragem clássica de Nyquist e de Bessel, bem como a respectiva ligação à interpolação de Lagrange e de Bessel. Na segunda parte, estudaremos a implementação numérica do método de amostragem por intermédio de funções q-Bessel (introduzido por D. Abreu [2]). O ramo da análise que estuda este tipo de funções é conhecido como q-cálculo. Foram implementadas as termos básicos, como sejam, o q-factorial (equivalente ao factorial clássico), o cálculo das funções q-Bessel, suas derivadas e respectivo método de amostragem. No final deste trabalho será dedicado a exemplos numéricos deste método.In this thesis we present Basic concepts of uniform and non-uniform sampling theory and their connection with interpolation theory. In the first part we will study the connection between the classic Nyquistsampling and Bessel-sampling and Lagrange- and Besselinterpolation. In the second part we study the numerical implementation of the sampling method using q-Bessel function introduced by D. Abreu [2]. To this end we implement basic terms of the q-Calculus, such as the q-factorial (equivalent to the classic factorial), the computation of q-Bessel functions and their derivatives, as well as the sampling method itself. In the end we present numerical examples for this method

Geometric Aspects of Frame Representations of Abelian Groups

We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.Comment: 20 pages; contact author: Eric Webe