Abstract
Установлены точные о ценки типа неравенст в Никольского для случ ая весовw a(x)=exp(−xa), x∈R, гдеa0. Точнее, еслиN n (a,p,q) — константа в формуле (2), то для многочленовР n степени п выполнено н еравенство∥P nwa∥p, ≦ CNn(a,p, q)∥Pnwa∥q. Далее, это — наилучшее неравенство такого в ида в том смысле, что существуе т последовательност ь многочленов {P n}, для к оторых выполнено и об ратное неравенство.
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References
G. Freud, On direct and converse theorems in the theory of weighted polynomial approximation,Math. Zeitschr.,126 (1972), 123–134.
G. Freud, On weightedL 1-approximation by polynomials,Studia Math.,46 (1973), 125–133.
G. Freud, On the theory of one-sided weightedL 1-approximation by polynomials, in “Linear Operators and Approximation, II”, ed. P. L. Butzer et al., ISNM, vol. 25, Birkhäuser Verlag (Basel, 1974), 285–303.
G. Freud, A. Giroux andQ. I. Rahman, Sur l'approximation polynomiale avec poids exp (− ¦x¦),Canad. J. Math.,30 (1978), 358–372.
A. L. Levin andD. S. Lubinsky, Canonical products and the weights exp (-|x|)α,α>1, with applications,J. Approx. Theory,49 (1987), 149–169.
A. L. Levin andD. S. Lubinsky, Weights on the real line that admit good relative polynomial approximation, with applications,J. Approx. Theory,49 (1987), 170–195.
B. Ja. Levin,Distribution of Zeros of Entire Functions, Transl. Math. Monograph, vol. 5, Amer. Math. Soc. (Providence, Rhode Island, 1964).
D. S. Lubinsky, Gaussian quadrature, weights on the whole real line and even entire functions with nonnegative even order derivatives,J. Approx. Theory,46 (1986), 297–313.
C. Markett, Nikolskii-type inequalities for Laguerre and Hermite expansions, in “Functions, Series, Operators” (Proc. Conf. Budapest, 1980), ed. J. Szabados, North Holland (Amsterdam, 1984), 811–834.
H. N. Mhaskar, Weighted analogues of Nikolskii-type inequalities and their applications, in “Conference in Honor of A. Zygmund”, vol. II, Wadsworth International (Belmont, 1983), 783–801.
H. N. Mhaskar, Weighted polynomial approximation,J. Approx. Theory,46 (1986), 100–110.
H. N. Mhaskar andE. B. Saff, Extremal problems for polynomials with exponential weights,Trans. Amer. Math. Soc.,285 (1984), 203–234.
P. Nevai, Orthogonal polynomials on the real line associated with the weight |x|α exp (-|x|α), I (in Russian),Acta Math. Acad. Sci. Hungar.,24 (1973), 335–342.
P. Nevai,Orthogonal Polynomials, Memoirs Amer. Math. Soc.213 (1979), 1–185.
P. Nevai, “Géza Freud, Orthogonal Polynomials and Christoffel Functions”,J. Approx. Theory,48 (1986), 3–167.
P. Nevai andV. Totik, Weighted polynomial inequalities,Constructive Approximation,2 (1986), 113–127.
E. B. Saff, Incomplete and orthogonal polynomials, in “Approximation Theory, IV”, ed. C. K. Chui et al., Academic Press (New York, 1983), 219–256.
R. A. Zalik, Inequalities for weighted polynomials,J. Approx. Theory,37 (1983), 137–146.
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This material is based upon research supported by the National Science Foundation under Grant No. DMS-84-19525, by the United States Information Agency under Senior Research Fulbright Grant No. 85-41612, and by the Hungarian Ministry of Education (first author). The work was started while the second author visited The Ohio State University between 1983 and 1985, and it was completed during the first author's visit to Hungary in 1985.
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Nevai, P., Totik, V. Sharp Nikolskii inequalities with exponential weights. Analysis Mathematica 13, 261–267 (1987). https://doi.org/10.1007/BF01909432
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DOI: https://doi.org/10.1007/BF01909432