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Banach Center Publications

1995 | 31 | 1 | 329-348
Tytuł artykułu

Asymptotic distribution of poles and zeros of best rational approximants to $x^α$ on [0,1]

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let $r_n* ∈ ℛ_{nn}$ be the best rational approximant to $f(x) = x^α$, 1 > α > 0, on [0,1] in the uniform norm. It is well known that all poles and zeros of $r_n*$ lie on the negative axis $ℝ_{<0}$. In the present paper we investigate the asymptotic distribution of these poles and zeros as n → ∞. In addition we determine the asymptotic distribution of the extreme points of the error function $e_n = f - r_n*$ on [0,1], and survey related convergence results.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
329-348
Opis fizyczny
Daty
wydano
1995
Twórcy
autor
• Institute for Constructive Mathematics, Department of Mathematics, University of South Florida, Tampa, Florida 33620, U.S.A.
autor
• TFH-Berlin/FB2, Luxemburgerstr. 10, 13353 Berlin 65, Germany
Bibliografia
• [Be1] S. Bernstein, Sur meilleure approximation de |x| par des polynômes de degrés donnés, Acta Math. 37 (1913), 1-57.
• [Be2] S. Bernstein, About the best approximation of $|x|^p$ by means of polynomials of very high degree, Bull. Acad. Sci. USRR, Cl. Math. Nat. Sci. 2 (1938), 169-190; also in: Collected Works, Vol. II, 262-272 (in Russian).
• [Bu1] A. P. Bulanov, Asymptotics for the least deviation of |x| from rational functions, Math. Sb. 76 (118) (1968), 288-303 (in Russian); English transl.: Math. USSR-Sb. 5 (1968), 275-290.
• [Bu2] A. P. Bulanov, The approximation of $x^{1/3}$ by rational functions, Vesci Akad. Navuk BSSR Ser. Fiz.-Navuk 1968, no. 2, 47-56 (in Russian).
• [FrSz] G. Freud and J. Szabados, Rational Approximation to $x^α$, Acta Math. Acad. Sci. Hungar. 18 (1967), 393-399.
• [Ga] T. Ganelius, Rational approximation of $x^α$ on [0,1], Anal. Math. 5 (1979), 19-33.
• [Go1] A. A. Gonchar, On the speed of rational approximation of continuous functions with characteristic singularities, Math. Sb. 73 (115) (1967), 630-638 (in Russian); English transl.: Math. USSR-Sb. 2 (1967).
• [Go2] A. A. Gonchar, Rational approximation of the function $x^α$, in: Constructive Theory of Functions (Proc. Internat. Conf., Varna 1970), Izdat. Bolgar. Akad. Nauk, Sofia, 1972, 51-53 (in Russian).
• [Go3] A. A. Gonchar, The rate of rational approximation and the property of single valuedness of an analytic function in a neighborhood of an isolated singular point, Mat. Sb. 94 (136) (1974), 265-282 (in Russian); English transl.: Math. USSR-Sb. 23 (1974).
• [Me] G. Meinardus, Approximation of Functions: Theory and Numerical Methods, Springer, New York 1967.
• [Ne] D. J. Newman, Rational approximation to |x|, Michigan Math. J. 11 (1964), 11-14.
• [Ri] T. J. Rivlin, An Introduction to the Approximation of Functions, Blaisdell, Waltham, Mass., 1969.
• [SaSt1] E. B. Saff and H. Stahl, Distribution of poles and zeros of best approximations for $|x|^α$, in: Constructive Theory of Functions 91, G. K. Ivanov et al. (eds.), Publ. of the Bulgarian Acad., Sofia, 1992, 249-257.
• [SaSt2] E. B. Saff and H. Stahl, Sequences in the Walsh table for $|x|^α$, J. Canad. Math. Soc., to appear.
• [St1] H. Stahl, Best uniform rational approximation of |x| on [-1,1], Mat. Sb. 183 (1992), 85-112 (in Russian); English transl.: Russian Acad. Sci. Sb. Math. 76 (1993), 461-487.
• [St2] H. Stahl, Uniform approximation of |x|, in: Methods of Approximation Theory in Complex Analysis and Mathematical Physics, A. A. Gonchar and E. B. Saff (eds.), Nauka, 1992, 110-130.
• [St3] H. Stahl, Best uniform rational approximation of $x^α$ on [0,1], Bull. Amer. Math. Soc. 28 (1993), 116-122.
• [St4] H. Stahl, Poles and zeros of best rational approximations of |x|, Constr. Approx. 10 (1994), 469-522.
• [StTo] H. Stahl and V. Totik, General Orthogonal Polynomials, Encyclopedia of Math. Appl. 43, Cambridge University Press, Cambridge, 1992.
• [VC] R. S. Varga and A. J. Carpenter, Some numeral results on best uniform rational approximation of $x^α$ on [0,1], Numer. Algorithms 2 (1992), 171-185.
• [Vy1] N. S. Vyacheslavov, The approximation of |x| by rational functions, Mat. Zametki 16 (1974), 163-171 (in Russian).
• [Vy2] N. S. Vyacheslavov, On the approximation of |x| by rational functions, Dokl. Akad. Nauk SSSR 220 (1975), 512-515; English transl.: Soviet Math. Dokl. 16 (1975), 100-104.
• [Vy3] N. S. Vyacheslavov, On the approximation of $x^α$ by rational functions, Izv. Akad. Nauk SSSR 44 (1980) (in Russian); English transl.: Math. USSR-Izv. 16 (1981), 83-101.
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