Abstract. 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.

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The goal of channel measurement or operator identification is to obtain complete knowledge of a channel operator by observing the image of a finite number of input signals. We shall show that if the spreading support of the operator (that is, the support of the symplectic Fourier transform of the Kohn-Nirenberg symbol of the operator) has area less than one then the operator is identifiable.If the spreading support is larger than one, then the operator is not identifiable. The shape of the support region is essentially arbitrary thereby proving a conjecture of Bello. The input signal considered is a weighted delta train where the weights are the window function of a finite Gabor system whose elements satisfy a certain robust completeness property.

ABSTRACT. We discuss the linear independence of systems of m vectors in

We develop an algorithm to reconstruct the wavelet coefficients of an image from the Radon transform data. The proposed method uses the properties of wavelets to localize the Radon transform and can be used to reconstruct a local region of the cross section of a body, using almost completely local data that significantly reduces the amount of exposure and computations in X-ray tomography. The property that distinguishes our algorithm from the previous algorithms is based on the observation that for some wavelet bases with sufficiently many vanishing moments, the ramp-filtered version of the scaling function as well as the wavelet function has extremely rapid decay. We show that the variance of the elements of the null-space is negligible in the locally reconstructed image. Also, we find an upper bound for the reconstruction error in terms of the amount of data used in the algorithm. To reconstruct a local region 16 pixels in radius in a 256x256 image, we require 22% of full exposure data.