Algebraic-polynomial approximation of functions satisfying a Lipschitz condition

Abstract

For functionsf(x) ε KH^(α) [satisfying the Lipschitz condition of order α (0 < α < 1) with constant K on [−1, 1], the existence is proved of a sequence P_n (f; x) of algebraic polynomials of degree n = 1, 2,..., such that

$$|f(x) - P_{n - 1} (f;x)| \leqslant \mathop {\sup }\limits_{f \in KH^{(\alpha )} } E_n (f)[(1 - x^2 )^{a/2} + o(1)],$$

when n → ∞, uniformly for x ε [−1, 1], where E_n(f) is the best approximation off(x) by polynomials of degree not higher than n.