  This paper presents some results on a well-known problem in Algebraic Signal
Sampling and in other areas of applied mathematics: reconstruction of
piecewise-smooth functions from their integral measurements (like moments,
Fourier coefficients, Radon transform, etc.). Our results concern
reconstruction (from the moments) of signals in two specific classes: linear
combinations of shifts of a given function, and "piecewise $D$-finite
functions" which satisfy on each continuity interval a linear differential
equation with polynomial coefficients.
