Oversampling in shift-invariant spaces with a rational sampling period

8 pages, no figures.It is well known that, under appropriate hypotheses, a sampling formula allows us to recover any function in a principal shift-invariant space from its samples taken with sampling period one. Whenever the generator of the shift-invariant space satisfies the Strang-Fix conditions of order r, this formula also provides an approximation scheme of order r valid for smooth functions. In this paper we obtain sampling formulas sharing the same features by using a rational sampling period less than one. With the use of this oversampling technique, there is not one but an infinite number of sampling formulas. Whenever the generator has compact support, among these formulas it is possible to find one whose associated reconstruction functions have also compact support.This work has been supported by the Grant MTM2009-08345 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología

El uso de bases y frames en teoría de muestreo

14 págs.-- El autor agradece la lectura crítica del trabajo por parte de los profesores Alberto Ibort (Universidad Carlos III de Madrid) y Gerardo Pérez-Villalón (Universidad Politécnica de Madrid).En este artículo se trata de la recuperación de x∈H, siendo H un espacio de Hilbert separable, a partir de la sucesión de escalares {}n=1,infty obtenida mediante una sucesión dada {xn}n=1,infty de H. El exigir que se cumpla el concepto de la l2-estabilidad nos lleva directamente a la definición de frame en H, siendo las bases ortonormales y las bases de Riesz dos casos particulares. Después de una breve excursión por las propiedades más importantes de los frames, y de su contrapartida en el caso de bases ortonormales y de bases de Riesz, nos centraremos en el caso en que H es un espacio de Hilbert de funciones y la sucesión {}n=1,infty consiste en muestras de x y/o de ciertas funciones relacionadas con ella. Como ilustración de la teoría anterior obtendremos teoremas de muestreo en los espacios clásicos de Paley–Wiener, así como en otros espacios invariantes por traslación en L2(R), i.e., subespacios cerrados de L2(R) generados por las traslaciones en los enteros de una cierta función φ de L2(R).Este trabajo ha sido financiado por el proyecto MTM2006-09737 de la D.G.I. del Ministerio de Ciencia y Tecnología

Universality in quantum chaos and the one parameter scaling theory

We adapt the one parameter scaling theory (OPT) to the context of quantum chaos. As a result we propose a more precise characterization of the universality classes associated to Wigner-Dyson and Poisson statistics which takes into account Anderson localization effects. Based also on the OPT we predict a new universality class in quantum chaos related to the metal-insulator transition and provide several examples. In low dimensions it is characterized by classical superdiffusion or a fractal spectrum, in higher dimensions it can also have a purely quantum origin as in the case of disordered systems. Our findings open the possibility of studying the metal insulator transition experimentally in a much broader type of systems.Comment: 4 pages, 2 figures, acknowledgment added, typos correcte

Introduction to nonlinearities, business cycles, and forecasting

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Is it possible to observe experimentally a metal-insulator transition in ultra cold atoms?

Kicked rotors with certain non-analytic potentials avoid dynamical localization and undergo a metal-insulator transition. We show that typical properties of this transition are still present as the non-analyticity is progressively smoothed out provided that the smoothing is less than a certain limiting value. We have identified a smoothing dependent time scale such that full dynamical localization is absent and the quantum momentum distribution develops power-law tails with anomalous decay exponents as in the case of a conductor at the metal-insulator transition. We discuss under what conditions these findings may be verified experimentally by using ultra cold atoms techniques. It is found that ultra-cold atoms can indeed be utilized for the experimental investigation of the metal-insulator transition.Comment: 7 pages, 3 figure

Effect of a magnetic flux on the critical behavior of a system with long range hopping

We study the effect of a magnetic flux in a 1D disordered wire with long range hopping. It is shown that this model is at the metal-insulator transition (MIT) for all disorder values and the spectral correlations are given by critical statistics. In the weak disorder regime a smooth transition between orthogonal and unitary symmetry is observed as the flux strength increases. By contrast, in the strong disorder regime the spectral correlations are almost flux independent. It is also conjectured that the two level correlation function for arbitrary flux is given by the dynamical density-density correlations of the Calogero-Sutherland (CS) model at finite temperature. Finally we describe the classical dynamics of the model and its relevance to quantum chaos.Comment: 5 pages, 4 figure