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A characterization is given of the sets supporting the uniform norms of weighted polynomials [w(x)]nPn(x), wherePn 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 andPn) 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.

Where does the 𝐿^{𝑝}-norm of a weighted polynomial live?