6 - Exact constants in Jackson inequalities

Summary

Jackson inequalities (or inequalities of Jackson's type) are expressions in which the approximation error of an individual function is estimated using the modulus of continuity of the function or some of its derivatives. In this context we can consider best approximations either from fixed subspaces or by given methods, and here in particular by linear ones. The only requirement for the approximating function is that the modulus used to estimate the error makes sense for it. The problem of obtaining inequalities which are exact on certain sets can also be posed.

The modulus of continuity is a better characteristic for a function than, for example, its norm in C or L_p and hence obtaining the exact constant in Jackson inequalities needs methods of investigation essentially different from those used in Chapters 4 and 5.

The proofs for exact inequalities of the Jackson type for polynomial and spline approximation are the main content of this chapter. The problem of finding the smallest constants in such inequalities with respect to the whole class of approximating subspaces of fixed dimension will be considered in Section 8.3.