Abstract
A characterization is given of the sets supporting the uniform norms of weighted polynomials [w(x)]n P n (x), whereP n is any polynomial of degree at mostn. The (closed) support ∑ ofw(x) may be bounded or unbounded; of special interest is the case whenw(x) has a nonempty zero setZ. 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 ofn andP 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.
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Communicated by Ronald A. DeVore.
Dedicated to Professor Geza Freud
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Mhaskar, H.N., Saff, E.B. Where does the sup norm of a weighted polynomial live?. Constr. Approx 1, 71–91 (1985). https://doi.org/10.1007/BF01890023
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DOI: https://doi.org/10.1007/BF01890023