Calculation of the Best Uniform Approximation in a Chebyshev System

In this chapter, the algorithms of de la Vallee Poussin and Remez are extended to generalized polynomials, which are functions generated by a Chebyshev system. The algorithm of de la Vallee Poussin provides the uniform approximation of a function by a generalized polynomial. The generalized Remez algorithm provides the calculation of the generalized polynomial with minimal uniform norm. In the chapter, the authors derive a result pertaining to the continuity of the best polynomial approximating scheme with respect to the function to be approximated. Both Borel-Chebyshev theorem and de la Vallee Poussin theorem indicate that the best uniform approximation ϕ* ∈ V of some function f ∈ C(0) ([a, b]) results as an approximation of f on a finite subset of points R; this set is called the characteristic set by Dzyadyk.