Introduction

Summary

This is the first in a series of books dealing with approximation and interpolation of functions. Many changes have occurred in this theory during the last decades. In what follows, we shall try to describe some of the problems and achievements of this period.

Until about 1955, the leading force in approximation was the Russians, in particular, Bernstein and his school (Ahiezer), Chebyshev, Kolmogorov, and Markov. The development of the subject in Germany, Hungary, and the United States occurred later. The West certainly leads in the number of papers published—see the bulky Journal of Approximation Theory. The twelve sections that follow review the newer developments.

The two classical books dealing with approximation and interpolation are those of Natanson [0–N] and Ahiezer [0–A]. Important recent books include two Russian works devoted to special problems: Korneichuk [0–K₂] (see also [0–K₃]) deals with best constants in the trigonometric approximation, while Tihomirov [0–T₁] treats extremal problems, particularly widths and optimization. The book of Butzer and Berens [0–B₂] introduced functional analytic methods into the field; the two books by de Boor [0–B₁] and Schumaker [0-S] deal with splines, an American development rich in practical applications. Karlin and Studden [0–K,] treat Chebyshev systems exhaustively. Books on general approximation theory are those of Rice [0–R], Lorentz [0–L], Dzyadyk [0–D], and Timan [0–T₂]; the last book contains a wealth of material. Several books will be mentioned in later sections.