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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Extremal problems for polynomials with exponential weights

Authors: H. N. Mhaskar and E. B. Saff
Journal: Trans. Amer. Math. Soc. 285 (1984), 203-234
MSC: Primary 41A17; Secondary 42C05
MathSciNet review: 748838
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Abstract: For the extremal problem:

$\displaystyle {E_{n,r}}(\alpha ): = \min \parallel \exp ( - \vert x{\vert^\alpha })\,({x^n} + \cdots ){\parallel_{{L^r}}}, \qquad \alpha > 0,$

where $ {L^r}\,(0 < r \leqslant \infty )$ denotes the usual integral norm over $ {\mathbf{R}}$, and the minimum is taken over all monic polynomials of degree $ n$, we describe the asymptotic form of the error $ {E_{n,r}}(\alpha )\;({\text{as}}\;n \to \infty )$ as well as the limiting distribution of the zeros of the corresponding extremal polynomials. The case $ r = 2$ yields new information regarding the polynomials $ \{ {p_n}(\alpha ;x) = {\gamma_n}(\alpha )\,{x^n} + \cdots \} $ which are orthonormal on $ {\mathbf{R}}$ with respect to $ \exp ( - 2\vert x{\vert^\alpha })$. In particular, it is shown that a conjecture of Freud concerning the leading coefficients $ {\gamma_n}(\alpha )$ is true in a Cesàro sense. Furthermore a contracted zero distribution theorem is proved which, unlike a previous result of Ullman, does not require the truth of the Freud's conjecture. For $ r = \infty ,\alpha > 0$ we also prove that, if $ \deg {P_n}(x) \leqslant n$, the norm $ \parallel \exp ( - \vert x\vert^{\alpha })\,{P_n}(x)\parallel_{{L^\infty }}$ is attained on the finite interval

$\displaystyle \left[ { - {{(n/{\lambda_\alpha })}^{1/\alpha }},{{(n/{\lambda_\a... ...ambda_\alpha } = \Gamma (\alpha )/{2^{\alpha - 2}}{\{ \Gamma (\alpha /2)\} ^2}.$

Extensions of Nikolskii-type inequalities are also given.

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

  • [1] Lars V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. MR 510197
  • [2] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, Vol. I, McGraw-Hill, New York, 1953.
  • [3] Paul Erdös and Paul Turán, On the uniformly-dense distribution of certain sequences of points, Ann. of Math. (2) 41 (1940), 162–173. MR 0001319
  • [4] G. Freud, Orthogonal polynomials, Pergamon Press, London, 1971.
  • [5] G. Freud, On two polynomial inequalities. I, Acta Math. Acad. Sci. Hungar. 22 (1971/1972), 109–116. MR 0288221
  • [6] Géza Freud, On the greatest zero of an orthogonal polynomial. II, Acta Sci. Math. (Szeged) 36 (1974), 49–54. MR 0346406
  • [7] G. Freud, On estimations of the greatest zeroes of orthogonal polynomials, Acta Math. Acad. Sci. Hungar. 25 (1974), 99–107. MR 0370043
  • [8] Géza Freud, On polynomial approximation with respect to general weights, Functional analysis and its applications (Internat. Conf., Eleventh Anniversary of Matscience, Madras, 1973; dedicated to Alladi Ramakrishnan), Springer, Berlin, 1974, pp. 149–179. Lecture Notes in Math., Vol. 399. MR 0404924
  • [9] Géza Freud, On the coefficients in the recursion formulae of orthogonal polynomials, Proc. Roy. Irish Acad. Sect. A 76 (1976), no. 1, 1–6. MR 0419895
  • [10] -, On the greatest zero of an orthogonal polynomial (preprint), Abstract 753-B36, Notices Amer. Math. Soc. 25 (1978).
  • [11] G. Freud, A. Giroux, and Q. I. Rahman, Sur l’approximation polynomiale avec poids 𝑒𝑥𝑝(-\mid𝑥\mid), Canad. J. Math. 30 (1978), no. 2, 358–372 (French). MR 0467115
  • [12] John Garnett, Analytic capacity and measure, Lecture Notes in Mathematics, Vol. 297, Springer-Verlag, Berlin-New York, 1972. MR 0454006
  • [13] G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
  • [14] N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. MR 0350027
  • [15] G. G. Lorentz, Approximation by incomplete polynomials (problems and results), Padé and rational approximation (Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976) Academic Press, New York, 1977, pp. 289–302. MR 0467089
  • [16] A. I. Markushevich, Theory of functions of a complex variable. Vol. III, Revised English edition, translated and edited by Richard A. Silverman, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0215964
  • [17] H. N. Mhaskar, Weighted polynomial approximation of entire functions. I, J. Approx. Theory 35 (1982), no. 3, 203–213. MR 663666, 10.1016/0021-9045(82)90002-8
  • [18] H. N. Mhaskar, Weighted polynomial approximation of entire functions. II, J. Approx. Theory 33 (1981), no. 1, 59–68. MR 639221, 10.1016/0021-9045(81)90089-7
  • [19] H. N. Mhaskar, Weighted analogues of Nikol′skiĭ-type inequalities and their applications, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 783–801. MR 730109
  • [20] -, On the convergence of expansions in polynomials orthogonal with respect to general weight functions on the whole real line, Acta Math. Acad. Sci. Hungar. (to appear).
  • [21] G. P. Névai, Polynomials that are orthogonal on the real axis with weight 𝑥^{𝛼}𝑒^{-𝑥^{𝛽}}. I, Acta Math. Acad. Sci. Hungar. 24 (1973), 335–342 (Russian). MR 0330559
  • [22] Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. MR 519926, 10.1090/memo/0213
  • [23] -, Orthogonal polynomials associated with $ \exp ( - {x^4})$, CMS Conference Proceedings, Vol. 3 (Second Edmonton Conference on Approximation Theory), Amer. Math. Soc., Providence, R.I., 1982, pp. 263-285.
  • [24] S. M. Nikol′skiĭ, Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of several variables, Trudy Mat. Inst. Steklov., v. 38, Trudy Mat. Inst. Steklov., v. 38, Izdat. Akad. Nauk SSSR, Moscow, 1951, pp. 244–278 (Russian). MR 0048565
  • [25] S. M. Nikol′skiĭ, Approximation of functions of several variables and imbedding theorems, Springer-Verlag, New York-Heidelberg., 1975. Translated from the Russian by John M. Danskin, Jr.; Die Grundlehren der Mathematischen Wissenschaften, Band 205. MR 0374877
  • [26] G. Pòlya and G. Szegö, Über den transfiniten Durchmesser (Kapäzitatskonstante) von ebenen und räumlichen Punktmengen, J. Reine Angew. Math. 165 (1931), 4-49.
  • [27] H. L. Royden, Real analysis, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1963. MR 0151555
  • [28] E. B. Saff, J. L. Ullman, and R. S. Varga, Incomplete polynomials: an electrostatics approach, Approximation theory, III (Proc. Conf., Univ. Texas, Austin, Tex., 1980), Academic Press, New York-London, 1980, pp. 769–782. MR 602801
  • [29] E. B. Saff and R. S. Varga, On incomplete polynomials. II, Pacific J. Math. 92 (1981), no. 1, 161–172. MR 618054
  • [30] G. Szegö, Orthogonal polynomials (3rd ed.), Amer. Math. Soc Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1967.
  • [31] A. F. Timan, Theory of approximation of functions of a real variable, Translated from the Russian by J. Berry. English translation edited and editorial preface by J. Cossar. International Series of Monographs in Pure and Applied Mathematics, Vol. 34, A Pergamon Press Book. The Macmillan Co., New York, 1963. MR 0192238
  • [32] J. L. Ullman, On the regular behaviour of orthogonal polynomials, Proc. London Math. Soc. (3) 24 (1972), 119–148. MR 0291718
  • [33] J. L. Ullman, Orthogonal polynomials associated with an infinite interval, Michigan Math. J. 27 (1980), no. 3, 353–363. MR 584699
  • [34] Joseph L. Ullman, On orthogonal polynomials associated with the infinite interval, Approximation theory, III (Proc. Conf., Univ. Texas, Austin, Tex., 1980), Academic Press, New York-London, 1980, pp. 889–895. MR 602816

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Keywords: Orthogonal polynomials, exponential weights, error estimates, zero distribution, Nikolskii inequalities
Article copyright: © Copyright 1984 American Mathematical Society