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)]nPn(x), where Pn is any polynomial of degree at most n. The (closed) support ∑ 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 ∑ -Z. One main result of this paper states that there is a unique compact set (independent of n and Pn) 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. © 1985 Springer-Verlag New York Inc.