  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 non-linear approaches. As a model
case we use piecewise-constant functions on [0,1], in particular, the Heaviside
jump function. Considered as a curve in the Hilbert space, 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 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.
