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We address in this paper the following two closely related problems: 1. How to represent functions with singularities (up to a prescribed accuracy) in a compact way? 2. How to reconstruct such functions from a small number of measurements? The stress is on a comparison of linear and nonlinear approaches. As a model case we use piecewise-constant functions on [0, 1], in particular, the Heaviside jump function H t = χ [0,t] . Considered as a curve in the Hilbert space L 2 ([0, 1]) it is completely characterized by the fact that any two its disjoint chords are orthogonal. We reinterpret this fact in a context of step-functions in one or two variables.Next we study the limitations on representability and reconstruction of piecewise-constant functions by linear and semi-linear methods. Our main tools in this problem are Kolmogorov's n-width and ǫ-entropy, as well as Temlyakov's (N, m)-width.On the positive side, we show that a very accurate non-linear reconstruction is possible. It goes through a solution of certain specific non-linear systems of algebraic equations. We discuss the form of these systems and methods of their solution, stressing their relation to Moment Theory and Complex Analysis.Finally, we informally discuss two problems in Computer Imaging which are parallel to the problems 1 and 2 above: compression of still images and video-sequences on one side, and image reconstruction from indirect measurement (for example, in Computer Tomography), on the other.----------------

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We consider the problem of "algebraic reconstruction" of linear combinations of shifts of several known signals f 1 , . . . , f k from the Fourier samples. Following [5], for each j = 1, . . . , k we choose sampling set S j to be a subset of the common set of zeroes of the Fourier transforms F(f ), = j, on which F(f j ) = 0. It was shown in [5] that in this way the reconstruction system is "decoupled" into k separate systems, each including only one of the signals f j . The resulting systems are of a "generalized Prony" form.However, the sampling sets as above may be non-uniform/not "dense enough" to allow for a unique reconstruction of the shifts and amplitudes.In the present paper we study uniqueness and robustness of non-uniform Fourier sampling of signals as above, investigating sampling of exponential polynomials with purely imaginary exponents. As the main tool we apply a well-known result in Harmonic Analysis: the Turán-Nazarov inequality ([18]), and its generalization to discrete sets, obtained in [12]. We illustrate our general approach with examples, and provide some simulation results.