Sampling and reconstruction of operators

We study the recovery of operators with bandlimited Kohn-Nirenberg symbol from the action of such operators on a weighted impulse train, a procedure we refer to as operator sampling. Kailath, and later Kozek and the authors have shown that operator sampling is possible if the symbol of the operator is bandlimited to a set with area less than one. In this paper we develop explicit reconstruction formulas for operator sampling that generalize reconstruction formulas for bandlimited functions. We give necessary and sufficient conditions on the sampling rate that depend on size and geometry of the bandlimiting set. Moreover, we show that under mild geometric conditions, classes of operators bandlimited to an unknown set of area less than one-half permit sampling and reconstruction. A similar result considering unknown sets of area less than one was independently achieved by Heckel and Boelcskei. Operators with bandlimited symbols have been used to model doubly dispersive communication channels with slowly-time-varying impulse response. The results in this paper are rooted in work by Bello and Kailath in the 1960s.Comment: Submitted to IEEE Transactions on Information Theor

Fundamental papers in wavelet theory

This book traces the prehistory and initial development of wavelet theory, a discipline that has had a profound impact on mathematics, physics, and engineering. Interchanges between these fields during the last fifteen years have led to a number of advances in applications such as image compression, turbulence, machine vision, radar, and earthquake prediction. This book contains the seminal papers that presented the ideas from which wavelet theory evolved, as well as those major papers that developed the theory into its current form. These papers originated in a variety of journals from diffe

Continuous And Discrete Wavelet Transforms

. This paper is an expository survey of results on integral representations and discrete sum expansions of functions in L 2 (R) in terms of coherent states. Two types of coherent states are considered: Weyl--Heisenberg coherent states, which arise from translations and modulations of a single function, and affine coherent states, called "wavelets," which arise as translations and dilations of a single function. In each case it is shown how to represent any function in L 2 (R) as a sum or integral of these states. Most of the paper is a survey of literature, most notably the work of I. Daubechies, A. Grossmann, and J. Morlet. A few results of the authors are included. Key words. frame, wavelet, coherent states, integral transform, Gabor transform, wavelet transform, Weyl--Heisenberg group, affine group AMS subject classifications. 42C15, 42A38 0. Introduction. The representation of a signal by means of its spectrum or Fourier transform is essential to solving many problems both in ..

Wavelets and Local Tomography

In this paper, formulas relating the Radon transform and Radon transform inversion to various wavelet and multiscale transformations, including the continuous wavelet transform, the semi--continuous wavelet transform of Mallat, steerable multiscale filters of Freeman and Adelson, and separable orthogonal and non--orthogonal wavelet bases, are given. The use of wavelets as a valuable tool in the local inversion of the Radon transform in even dimensions is justified, and explicit estimates on the decay of ramp-- filtered wavelets is given. 1 Introduction The problem of local tomography can be stated as follows. Given a ? 0, compute the values of a function f(x), x 2 R n , for all x such that jxj a, from knowledge of the projections of f on lines passing through the ball of radius a about the origin. In other language, the problem of local tomography is the recovery of the function f(x) fjxjag (x) from R ` f(s) [\Gammaa;a] (s), where S (t) = 1 if t is in the set S, and S (t..