Markov–bernstein Type Inequalities on Compact Subsets of R

play an important role in approximation theory. Various generalizations in several directions are well-known; for a survey of these results see the recent monograph of P. Borwein and T. Erdélyi [2]. A number of papers study possible extensions of Markov-type inequalities to compact sets K ⊂ R when the geometry of K is known apriori (Cantor type sets, finitely many intervals, etc.; cf. W. Pleśniak [7] and the references therein, as well as Borwein and Erdélyi [1], Totik [9], [10]), and this determines the approach to the above mentioned inequalities. In this paper we consider another possible path of generalizations. Instead of the knowledge of the geometry of the set, we define some density functions of the set in the neighborhood of a point, and estimate the derivative at this point. With this approach we will be able to settle the problem for many interesting sets, and when this method breaks down then we use an interpolation theoretic approach. In both situations, we distinguish Markov type inequalities (when we use information about the polynomial only on one side of the point), and Bernstein type inequalities (when information is provided on both sides of the point). Although our results are formulated for one point, with a proper modification of the density functions we could establish uniform estimates on the whole set.