Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials)
A characterization is given of the sets supporting the uniform norms of weighted polynomials \([w(x)]^ nP_ n(x)\), where \(P_ n\) is any polynomial of degree at most n. The (closed) support \(\Sigma\) of w(x) may be bounded or unbounded; of special interest is the case when w(x) has a nonempty zero set Z. The treatment of weighted polynomials consists of associating each admissible weight with a certain functional defined on subsets of \(\Sigma\) \(\setminus Z.\) One main result of this paper states that there is a unique compact set (independent of n and \(P_ n)\) maximizing this functional that contains the points where the norms of weighted polynomials are attained. The distribution of the zeros of Chebyshev polynomials corresponding to the weights \([w(x)]^ n\) is also studied. The main theorems give a unified method of investigating many particular examples. Applications to weighted approximation on the real line with respect to a fixed weight are included.
distribution of the zeros of Chebyshev polynomials
scientific article; zbMATH DE number 3931644
scientific article