Characterization of approximation schemes satisfying Shapiro's Theorem

TL;DR

This paper characterizes approximation schemes that satisfy Shapiro's theorem, applying the results to classical approximation processes, and compares with prior results, highlighting schemes that do not satisfy the theorem.

Contribution

It provides a comprehensive characterization of schemes satisfying Shapiro's theorem and explores their application to various classical approximation methods.

Findings

Characterization of approximation schemes satisfying Shapiro's theorem

Application to finite rank and n-term approximation processes

Identification of schemes that do not satisfy Shapiro's theorem

Abstract

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.