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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
DOI: https://doi.org/10.1090/S0002-9947-1987-0896010-9
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.

References [Enhancements On Off] (What's this?)

  • [1] S. S. Bonan, Weighted mean convergence of Largrange interpolation, Ph.D. Dissertation, Ohio State Univ., Columbus, Ohio, 1982.
  • [2] G. Freud, Orthogonal polynomials, Pergamon Press, London, 1971.
  • [3] -, On polynomial approximation with respect to general weights, Lecture Notes in Math., vol. 399, Springer-Verlag, Berlin and New York, 1974, pp. 149-179. MR 0404924 (53:8722)
  • [4] -, On Markov-Berstein-type inequalities and their applications, J. Approximation Theory 35 (1982), 203-213.
  • [5] A. A. Gonchar and E. A. Rakhmanov, The equilibrium measure and distribution of zeros of extremal polynomials, Math. Sb. 125 (167) (1984), 117-127. MR 760416 (86f:41002)
  • [6] N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, Berlin and New York, 1972. MR 0350027 (50:2520)
  • [7] D. S. Lubinsky, A weighted polynomial inequality, Proc. Amer. Math. Soc. 92 (1984), 263-267. MR 754716 (86e:41021)
  • [8] H. N. Mhaskar, A rate of convergence theorem for expansions in Freud polynomials, T. R. #8608, Bowling Green State Univ., 1986.
  • [9] H. N. Mhaskar and E. B. Saff, Extremal problems for polynomials with exponential weights, Trans. Amer. Math. Soc. 285 (1984), 203-234. MR 748838 (86b:41024)
  • [10] -, Polynomials with Laguerre weights in $ {L^p}$, Lecture Notes in Math., vol. 1105, Springer-Verlag, Berlin and New York, 1984, pp. 511-523.
  • [11] -, Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials), Constr. Approx. 1 (1985), 71-91. MR 766096 (86a:41004)
  • [12] -, A Weierstrass-type theorem for certain weighted polynomials, Proc. Internat. Conf. on Approximation Theory, St. Johns, N.F., Canada, 1984 (to appear).
  • [13] P. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. No. 213 (1979). MR 519926 (80k:42025)
  • [14] M. Tsuji, Potential theory in modern function theory, 2nd ed., Chelsea, New York, 1958. MR 0114894 (22:5712)
  • [15] R. A. Zalik, Some weighted polynomial inequalities, J. Approximation Theory 41 (1984), 39-50. MR 742234 (85k:41021)
  • [16] A. Zygmund, Trigonometric series, Cambridge Univ. Press, Cambridge, 1977.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1987-0896010-9
Keywords: Weighted polynomials, orthogonal polynomials, extremal polynomials, Nikolskii-type inequalities, potential theory
Article copyright: © Copyright 1987 American Mathematical Society

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