Weighted polynomials on discrete sets (Q1871766)

Let \(I\) be a real interval of positive length. A weighted polynomial of degree at most \(n\geq 1\) on \(I\) is an expression of the form \(P_nw^n\) where \(P_n\) is an algebraic polynomial of degree at most \(n\geq 1\) and \(w:I\rightarrow [0,\infty)\) is a positive, not identically zero continuous weight on \(I\). In this paper, the author proves a necessary and sufficient condition which ensures that the continuous \(L_p\) (\(0<p\leq \infty\)) norm of a weighted polynomial, \(P_nw^n\), \(\deg P_n\leq n\), \(n\geq 1,\) is in an \(n\)th root sense, controlled by its corresponding discrete Hölder norm on a very general class of discrete subsets of \(I\). As a consequence, he establishes Nikol'skij inequalities and theorems dealing with zero distribution, zero location and sup and \(L_p\) infinite-finite range inequalities.