Extremal Problems of Approximation Theory

The main subject of this chapter is the study of best approximation either to separate functions or to classes of functions by polynomials of given degree as well as by approximants from other subspaces. In Section 5.1, we not only outline, in subsections A, B, and C, the precise setting of the investigated problems but also discuss questions of existence and uniqueness of best approximation and give a criterion of best approximation. In Section 5.2 we introduce a specific definition for the space L p (Ω, μ) , while for C on the compact the same procedure is done in Section 5.3. In Section 5.5 we discuss best approximation to classes of functions by polynomials and by entire functions of exponential type. In Section 5.4 extremal properties of splines (5.4.9 – 5.4.12) are given. We are going to use them further on, in Chapter 10. We also study the properties of polynomials (5.4.1 – 5.4.8 and 5.4.13 – 5.4.14) concerning best approximation to a constant by algebraic polynomials with integral coefficients (5.4.15 – 5.4.16).