Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials). (English) Zbl 0582.41009

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.


41A10 Approximation by polynomials
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
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