Direct and inverse theorems of approximation

Summary

Direct and inverse theorems are the two halves of the link between the smoothness of a function and how fast it can be approximated. A direct theorem, of Jackson type, says that smooth functions have small errors of best approximation. An inverse theorem, of Bernstein type, says that small errors of best approximation force smoothness. Together they often characterise a smoothness class exactly by the rate of approximation.