Let [-1,1] be the space of continuous functions on [-,1], and denote by Δ2 the set of convex functions f [-,1]. Also, let E n (f) and E n (2) (f) denote the degrees of best unconstrained and convex approximation of f Δ2 by algebraic polynomials of degree < n, respectively. Clearly, En (f) E n (2) (f), and Lorentz and Zeller proved that the inverse inequality E n (2) (f) cE n (f) is invalid even with the constant c = c(f) which depends on the function f Δ2. In this paper we prove, for every α > 0 and function f Δ2, that ⊃ α En(2) (f):n q c(α)⊃ α En (f):n N, where c(α) is a constant depending only on α. Validity of similar results for the class of piecewise convex functions having s convexity changes inside (-1,1) is also investigated. It turns out that there are substantial differences between the cases 1 and s 2.
- (co)convex approximation by polynomials
- Degree of approximation