Direct and converse theorems of approximation theory and semigroups of operators

One of the important aims of the modem constructive theory of functions is to establish relationships between the structural properties 6i functions and sequences of approximations to them. The foundations of work in this field were laid by Jackson, Bernstein, and de la Vallée-Poussin. Subsequent developments were made by Zygmund, Kolmogorov, Nikol'skii, Pavard, and others.
Jackson' s classical inequality and the fundamental converse theorem of Bernstein-de la Vallée–Poussin, which were initially established for approximations to continuous functions by algebraic and trigonometric polynomials, have been generalized in various directions. Direct and converse theorems have been obtained for algebraic and trigonometric approximations in spaces other than $C$, for spaces of almost periodic functions, for approximations by eigenfunctions of a Sturm–Liouville problem, and so on.
The purpose of the present paper is to set forth the basic direct and converse theorems of the theory of approximations in Banach spaces. The main technique of the investigation is the use of strongly continuous semigroups of operators and the resolvents of operators generating these semigroups. Under certain conditions on the resolvent (see Ch. II, § 1), general direct and converse theorems are established for approximations by eigen-subspaces of a generating operator. These general theorems include as special cases many of the previously known results in the constructive theory of functions.