Uniform Lipschitz constants on small intervals

AbstractLet f in C[− 1, 1] be given, and let n be a fixed nonnegative integer. For 0 & θ ⩽ 1 define Pθ(f) to be the polynomial of degree less than or equal to n of best uniform approximation to f on [−θ, θ]. It is well known that there exists for each such θ, a constant λf(θ) such that for all g ϵ C[ − θ, θ], ∥ Pθ− Pθ(g) ∥[− θ,θ ⩽ λf(θ) ∥. Sufficient conditions on f are obtained to ensure that the set {λf(θ)|0 & θ ⩽ γ} is bounded for some δ > 0. An example is given showing that {λf(θ)|0 & θ ⩽ γ} may be bounded for some δ < 1 but not for δ − 1