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.

MSC:

 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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