  In this paper we characterize the approximation schemes that satisfy
Shapiro's theorem and we use this result for several classical approximation
processes. In particular, we study approximation of operators by finite rank
operators and n-term approximation for several dictionaries and norms.
Moreover, we compare our main theorem with a classical result by Yu. Brundyi
and we show two examples of approximation schemes that do not satisfy Shapiro's
theorem.
