Inequalities and Inverse Theorems in Restricted Rational Approximation Theory

Abstract

The following lemma, part a) due to S. N. Bernstein and part b) due to A. A. Markov, is fundamental to the proofs of many inverse theorems in polynomial approximation theory.LEMMA 1. [2, p. 62 and p. 67] Let ∏n denote the real polynomials of degree at most n. Let p ∈ ∏n, thena)b)From the lemma one can deduce, for example:THEOREM 1. Iƒ there is a sequence of polynomials pn ∈ ∏n and a δ > 0 so that ‖f – pn‖[a, b] ≧ A/nk + δ then f is k times continuously differentiate on (a, b).We shall refer to inequalities, such as those of Lemma 1 that bound the derivative r′ of a rational function of degree n in terms of its supremum norm ‖r‖[a, b] and n, as Bernstein-type inequalities.