BERNSTEIN TYPE THEOREMS FOR COMPACT-SETS IN R(N)

In this paper we give two non-trivial generalizations of a classical Bernstein inequality which is apparently less known that that of Bernstein-Markov, viz. |p′(x)| ≤ k(1 - x2) -1 2 (∥p∥[-1, 1]2 - p2(x)) 1 2, for x ε{lunate} (-1, 1), where p is a real polynomial of deg p ≤ k and ∥p∥[-1, 1] = sup|p|([-1, 1]), to the case of a compact set E in Rn with nonempty interior. Contrary to the situation where estimates for p′(x) are sought on the whole compact set, we do not, in general, need any other assumptions on E. Our results point out connections between Bernstein's inequality and two important notions in modern polynomial approximation theory on compacta in Cn: Siciak's extremal function and complex equilibrium measure. © 1992.