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.
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 |
Keywords:
characterization; weighted polynomials; distribution of the zeros of Chebyshev polynomials; weighted approximationReferences:
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