Stability of the Recovery of Missing Samples in Derivative Oversampling

This paper deals with the problem of reconstructing a band-limited signal when a finite subset of its samples and of its derivative are missing. The technique used, due to P.J.S.G. Ferreira, is based on the use of a particular frame for band-limited functions and the relative oversampling formulas. We study the eigenvalues of the matrices arising in the procedure of recovering the lost samples, finding estimates of their eigenvalues and their dependence on the oversampling parameter and on the number of missing samples. When the missing samples are consecutive, the problem may become very ill-conditioned. We present a numerical procedure to overcome this difficulty, also in presence of noisy data, by using Tikhonov regularization techniques.Comment: 17 pages,8 figure

Recovery of Missing Samples in Oversampling Formulas for Band Limited Functions

In a previous paper, the author constructed frames and oversampling formulas for band-limited functions, in the framework of the theory of shift-invariant spaces. In this article we study the problem of recovering missing samples. We find a sufficient condition for the recovery of a finite set of missing samples. The condition is expressed as a linear independence of the components of a vector W over the space of trigonometric polynomials determined by the frequencies of the missing samples. We apply the theory to the derivative sampling of any order and we illustrate our results with a numerical experiment.Comment: 19 pages, 3 figures, corrected a few typo

NUMERICAL SIMULATION OF VORTEX BREAKDOWN

Vortex breakdown is simulated by a three dimensional Lagrangian method using vortex filaments. The filaments are approximated by vortex elements and their velocity is computed by a Biot-Savart type law of interaction. The numerical calculations show the development of an axisymmetric bubble with a recirculation zone and resemble in many respects the results obtained in the physical experiments on vortex breakdown

On a necessary condition for B-spline Gabor frames

In a previous note K. Grochenig et al. prove that if g is a continuous function with compact support such that the translates of g form a partition of unity, then g cannot generate a Gabor frame for integer values of the frequency shift parameter b greater than 1. We give a simpler proof of this result which applies also to windows g which are neither continuous nor with compact support. Our proof is based on a necessary condition for Gabor frames due to C. E. Heil and D. F. Walnut

NUMERICAL SIMULATION OF VORTEX BREAKDOWN

Vortex breakdown is simulated by a three dimensional Lagrangian method using vortex filaments. The filaments are approximated by vortex elements and their velocity is computed by a Biot-Savart type law of interaction. The numerical calculations show the development of an axisymmetric bubble with a recirculation zone and resemble in many respects the results obtained in the physical experiments on vortex breakdown

Frames and oversampling formulas for band limited functions

In this article, we obtain families of frames for the space B (omega) of functions with band in [-omega, omega] by using the theory of shift-invariant spaces. Our results are based on the Gramian analysis of Ron and Shen and a variant, due to Bownik, of their characterization of families of functions whose shifts form frames or Riesz bases. We give necessary and sufficient conditions for the translates of a finite number of functions (generators) to be a frame or a Riesz basis for B (omega) . We also give explicit formulas for the dual generators, and we apply them to Hilbert transform sampling and derivative sampling. Finally we provide numerical experiments that support the theory