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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Where does the $ L\sp p$-norm of a weighted polynomial live?


Authors: H. N. Mhaskar and E. B. Saff
Journal: Trans. Amer. Math. Soc. 303 (1987), 109-124
MSC: Primary 41A65; Secondary 42C10
Erratum: Trans. Amer. Math. Soc. 308 (1988), 431.
MathSciNet review: 896010
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Abstract: For a general class of nonnegative weight functions $ w(x)$ having bounded or unbounded support $ \Sigma \subset {\mathbf{R}}$, the authors have previously characterized the smallest compact set $ {\mathfrak{S}_w}$, having the property that for every $ n = 1,\,2, \ldots $ and every polynomial $ P$ of degree $ \leqslant n$,

$\displaystyle \vert\vert{[w(x)]^n}P(x)\vert{\vert _{{L^\infty }(\Sigma )}} = \vert\vert{[w(x)]^n}P(x)\vert{\vert _{{L^\infty }({\mathfrak{S}_w})}}$

. In the present paper we prove that, under mild conditions on $ w$, the $ {L^p}$-norms $ (0 < p < \infty )$ of such weighted polynomials also "live" on $ {\mathfrak{S}_w}$ in the sense that for each $ \eta > 0$ there exist a compact set $ \Delta $ with Lebesgue measure $ m(\Delta ) < \eta $ and positive constants $ {c_1}$, $ {c_2}$ such that

$\displaystyle \vert\vert{w^n}P\vert{\vert _{{L^p}(\Sigma )}} \leqslant (1 + {c_... ... - {c_2}n))\vert\vert{w^n}P\vert{\vert _{{L^p}({\mathfrak{S}_w} \cup \Delta )}}$

. As applications we deduce asymptotic properties of certain extremal polynomials that include polynomials orthogonal with respect to a fixed weight over an unbounded interval. Our proofs utilize potential theoretic arguments along with Nikolskii-type inequalities.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0896010-9
PII: S 0002-9947(1987)0896010-9
Keywords: Weighted polynomials, orthogonal polynomials, extremal polynomials, Nikolskii-type inequalities, potential theory
Article copyright: © Copyright 1987 American Mathematical Society