Where does the norm of a weighted polynomial live?
Authors:
H. N. Mhaskar and E. B. Saff
Journal:
Trans. Amer. Math. Soc. 303 (1987), 109124
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 having bounded or unbounded support , the authors have previously characterized the smallest compact set , having the property that for every and every polynomial of degree , . In the present paper we prove that, under mild conditions on , the norms of such weighted polynomials also "live" on in the sense that for each there exist a compact set with Lebesgue measure and positive constants , such that . 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 Nikolskiitype inequalities.
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 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, SpringerVerlag, Berlin and New York, 1974, pp. 149179. MR 0404924 (53:8722)
 [4]
 , On MarkovBersteintype inequalities and their applications, J. Approximation Theory 35 (1982), 203213.
 [5]
 A. A. Gonchar and E. A. Rakhmanov, The equilibrium measure and distribution of zeros of extremal polynomials, Math. Sb. 125 (167) (1984), 117127. MR 760416 (86f:41002)
 [6]
 N. S. Landkof, Foundations of modern potential theory, SpringerVerlag, Berlin and New York, 1972. MR 0350027 (50:2520)
 [7]
 D. S. Lubinsky, A weighted polynomial inequality, Proc. Amer. Math. Soc. 92 (1984), 263267. 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), 203234. MR 748838 (86b:41024)
 [10]
 , Polynomials with Laguerre weights in , Lecture Notes in Math., vol. 1105, SpringerVerlag, Berlin and New York, 1984, pp. 511523.
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 , Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials), Constr. Approx. 1 (1985), 7191. MR 766096 (86a:41004)
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 , A Weierstrasstype theorem for certain weighted polynomials, Proc. Internat. Conf. on Approximation Theory, St. Johns, N.F., Canada, 1984 (to appear).
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 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), 3950. MR 742234 (85k:41021)
 [16]
 A. Zygmund, Trigonometric series, Cambridge Univ. Press, Cambridge, 1977.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198708960109
PII:
S 00029947(1987)08960109
Keywords:
Weighted polynomials,
orthogonal polynomials,
extremal polynomials,
Nikolskiitype inequalities,
potential theory
Article copyright:
© Copyright 1987
American Mathematical Society
