This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of band-limited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shift-invariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) prefilters that are not necessarily ideal low-pass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Band-limited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,

We present a new “second generation” reconstruction algorithm for irregular sampling, i.e. for the problem of recovering a band-limited function from its non-uniformly sampled values. The efficient new method is a combination of the adaptive weights method which was developed by the two first named authors and the method of conjugate gradients for the solution of positive definite linear systems. The choice of ”adaptive weights” can be seen as a simple but very efficient method of preconditioning. Further substantial acceleration is achieved by utilizing the Toeplitztype structure of the system matrix. This new algorithm can handle problems of much larger dimension and condition number than have been accessible so far. Furthermore, if some gaps between samples are large, then the algorithm can still be used as a very efficient extrapolation method across the gaps.

We introduce a new concept to describe the localization of frames. In our main result we shown that the frame operator preserves this localization and that the dual frame possesses the same localization property. As an application we show that certain frames for Hilbert spaces extend automatically to Banach frames. Using this abstract theory, we derive new results on the construction of nonuniform Gabor frames and solve a problem about non-uniform sampling in shift-invariant spaces. 1.

It is shown that a function f 2 L p [\GammaR; R], 1 p ! 1, is completely determined by the samples of f on sets = [ m i=1 fn=2r i g n2Z where R = P r i , and r i =r j is irrational if i 6= j, and of f (j) (0) for j = 1; : : : ; m \Gamma 1. If f 2 C m\Gamma2\Gammak [\GammaR; R], then the samples of f on and only the first k derivatives of f at 0 are required to completely determine f . Higher dimensional analogues of these results, which apply to functions f 2 L p [\GammaR; R] d and C m\Gamma2\Gammak [\GammaR; R] d , are proven. The sampling results are sharp in the sense that if any condition is omitted, there exist nonzero f 2 L p [\GammaR; R] d and C m\Gamma2\Gammak [\GammaR; R] d satisfying the rest. It is shown that the one dimensional sampling sets correspond to Bessel sequences of complex exponentials which are not Riesz bases for L 2 [\GammaR; R]. A signal processing application in which such sampling sets arise naturally is described in deta...

. We will report on significant improvements of some of the algorithms (cf. [12]) for reconstruction of band-limited signals from nonuniform sampling sets. Although the methods apply in principle to any dimension we shall put special emphasis in this report on the application to band-limited images. Especially the combination of the ideas of the Adaptive Weights Method suggested by Feichtinger and Grochenig with the conjugate gradient approach is better than the methods described so far in the literature, sometimes by several orders of magnitude. There is also experimental evidence that a combination of the steepest descent method with the use of a suitable relaxation parameter gives very often a highly efficient reconstruction method. In case of product sampled images successive 1D reconstruction performs best. Several examples illustrate the results. 1 Introduction We deal with the following discrete situation. A band-limited image is considered, i.e. a rectangular matr...

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. Two standard tools for signal analysis are the short--time Fourier transform and the continuous wavelet transform. These tools arise as matrix coefficients of square integrable representations of the Heisenberg and affine groups respectively, and discrete frame decompositions of L 2 arise from approximations of corresponding reproducing formulae. Here we study two groups, the so-called affine Weyl-Heisenberg and upper triangular groups, which contain both affine and Heisenberg subgroups. Generalized notions of square--integrable group representations allow us to fashion frames for L 2 and other function spaces. Such frames combine advantages of the short-time Fourier transform and wavelet transform and can be tailored to analyze specific types of signals. 1 Research supported by an Australian Research Council grant. 2 Research supported by NSF contract DMS-9307655. Typeset by A M S-T E X 1. INTRODUCTION. Wavelet and Gabor techniques have played a prominent role in both signa...

sampling in chirp space