We shall begin with a short survey of the basic results related to linear approximations (i.e. approximation by means of linear subspaces) so that one can feel better the peculiarities, the advantages as well as some shortcomings of the rational approximation. In this chapter we shall consider the problems of existence, uniqueness and characterization of the best approximation (best polynomial approximation). At the end of the chapter we shall consider also numerical algorithms for finding the best uniform polynomial approximation.

Approximation in normed linear spaces

Let X be a normed linear space. Recall that X is said to be a normed linear space if:

1. (i) X is a linear space, i.e. for its elements sum, and product with real numbers, are defined so that the standard axioms of commutativity and associativity are satisfied;

2. (ii) X is a normed space, i.e. to each x ϵ X there corresponds a nonnegative real number ∥x∥ satisfying the axioms

1. (a) ∥x∥ ≥ 0, ∥x∥ = 0 iff x = 0,

2. (b) ∥λx∥ = |λ| ∥x∥, λ a real number,

3. (c) ∥x + y∥ ≤ ∥x∥ + ∥y∥ (the triangle inequality).