  Many reconstruction problems in signal processing require solution of a
certain kind of nonlinear systems of algebraic equations, which we call Prony
systems. We study these systems from a general perspective, addressing
questions of global solvability and stable inversion. Of special interest are
the so-called "near-singular" situations, such as a collision of two closely
spaced nodes.
  We also discuss the problem of reconstructing piecewise-smooth functions from
their Fourier coefficients, which is easily reduced by a well-known method of
K.Eckhoff to solving a particular Prony system. As we show in the paper, it
turns out that a modification of this highly nonlinear method can reconstruct
the jump locations and magnitudes of such functions, as well as the pointwise
values between the jumps, with the maximal possible accuracy.
