Weighted Polynomials on Discrete Sets

Abstract.

 For a real interval I of positive length, we prove a necessary and sufficient condition which ensures that the continuous L _p (0 < p ⩽ ∞) norm of a weighted polynomial, P _n w ⁿ, deg P _n  ⩽ n, n ⩾ 1 is in an nth root sense, controlled by its corresponding discrete Hölder norm on a very general class of discrete subsets of I. As a by product of our main result, we establish inequalities and theorems dealing with zero distribution, zero location and sup and L _p infinite–finite range inequalities.