Markov and Bernstein Type Inequalities on Subsets

We extend Markov’s, Bernsteins’s, and Videnskii’s inequalities to arbitrary subsets of [−1, 1] and [−π,π], respectively. The primary purpose of this note is to extend Markov’s and Bernsteins’s inequalities to arbitrary subsets of [−1, 1] and [−π, π], respectively. We denote by Pn the set of all real algebraic polynomials of degree at most n and let m(·) denote the Lebesque measure of a subset of R. We were led to the results of this paper by the following problem. Can one give polynomials pn ∈ Pn and numbers an ∈ (0, 1), n = 1, 2, · · · , such that (i) m({x ∈ [0, 1] : |pn(x)| ≤ 1}) ≥ 1 − an, (ii) max 0≤x≤an |pn(x)| ≤ 1 and (iii) lim n→∞ n |pn(0)| = ∞ are satisfied? This question was asked by Vilmos Totik, and a positive answer would have been used in proving a conjecture in the theory of orthogonal polynomials. However, Theorem 2 of this note shows that the answer to the above question is negative, in fact, it gives slightly more. In addition, our Theorem 1 answers the corresponding question for trigonometric polynomials. Though our results cannot be used for Totik’s original purpose, our proofs illustrate well, how Remez-type inequalities can be used in proving various other polynomial inequalities. In this note we prove the following pair of theorems. Theorem 1. Let 0 < a ≤ 2π, 0 < L ≤ 1, let A be a closed subset of [0, 2π] with Lebesque measure m(A) ≥ 2π − a. There is an absolute constant c1 > 0 such that max t∈I |p′(t)| ≤ c1L−1(n + na) max t∈A |p(t)| (1) for every real trigonometric polynomial p of degree at most n and for every subinterval I of A with length at least La. This material is based upon work supported by the National Science and Engineering Research Council of Canada (P.B.) and the National Science Foundation under Grant No. DMS-9024901 (T.E.)