Algebraic reconstruction of piecewise-smooth functions from integral measurements

摘要

This paper presents some results on a well-known problem in Algebraic SignalSampling and in other areas of applied mathematics: reconstruction ofpiecewise-smooth functions from their integral measurements (like moments,Fourier coefficients, Radon transform, etc.). Our results concernreconstruction (from the moments or Fourier coefficients) of signals in twospecific classes: linear combinations of shifts of a given function, and"piecewise $D$-finite functions" which satisfy on each continuity interval alinear differential equation with polynomial coefficients. In each case theproblem is reduced to a solution of a certain type of non-linear algebraicsystem of equations ("Prony-type system"). We recall some known methods forexplicitly solving such systems in one variable, and provide extensions to somemulti-dimensional cases. Finally, we investigate the local stability of solvingthe Prony-type systems.