Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 

 

Best uniform rational approximation of $ x^\alpha$ on $ [0,1]$


Author: Herbert Stahl
Journal: Bull. Amer. Math. Soc. 28 (1993), 116-122
MSC: Primary 41A20
MathSciNet review: 1168517
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A strong error estimate for the uniform rational approximation of $ {x^{\alpha}}$ on [0, 1] is given, and its proof is sketched. Let $ {E_{nn}}({x^\alpha },[0,1])$ denote the minimal approximation error in the uniform norm. Then it is shown that

$\displaystyle \mathop {\lim }\limits_{x \to \infty } {e^{2\pi \sqrt {\alpha n} }}{E_{nn}}({x^\alpha },[0,1]) = {4^{1 + \alpha }}\vert\sin \pi \alpha \vert$

holds true for each $ {\alpha > 0}$.

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

  • [Be1] S. Bernstein, Sur meilleure approximation de $ {\vert x\vert}$ par des polynômes de degrés donnés, Acta Math. 37 (1913), 1-57.
  • [Be2] -, About the best approximation of $ {\vert x\vert^{p}}$ by means of polynomials of very high degree, Bull. Acad. Sci. USSR Cl. Sci. Math. Natur. 2 (1938), 169-190; also Collected Works, vol. II, 262-272. (Russian)
  • [Bu1] A. P. Bulanov, Asymptotics for least deviations of 𝑥 from rational functions, Mat. Sb. (N.S.) 76 (118) (1968), 288–303 (Russian). MR 0228889
  • [Bu2] A. P. Bulanov, The approximation of 𝑥^{1/3} by rational functions, Vesci Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk 1968 (1968), no. 2, 47–56 (Russian). MR 0231094
  • [FrSz] G. Freud and J. Szabados, Rational approximation to 𝑥^{𝛼}, Acta Math. Acad. Sci. Hungar 18 (1967), 393–399. MR 0221169
  • [Ga] T. Ganelius, Rational approximation to 𝑥^{𝛼} on [0,1], Anal. Math. 5 (1979), no. 1, 19–33 (English, with Russian summary). MR 535494, 10.1007/BF02079347
  • [GoLa] A. H. Gonchar and G. Lopez, On Markov's theorem for multipoint Padeé approximants, Mat. Sb. 105 (1978); English transl. in Math. USSR Sb. 34 (1978), 449-459.
  • [Go1] A. A. Gončar, Rapidity of rational approximation of continuous functions with characteristic singularities, Mat. Sb. (N.S.) 73 (115) (1967), 630–638 (Russian). MR 0214982
  • [Go2] -, Rational approximation of the function $ {x^{\alpha}}$, Constructive Theory of Functions (Proc. Internat. Conf., Varna 1970), Izdat. Bolgar. Akad. Nauk, Sofia, 1972, pp. 51-53. (Russian)
  • [Go3] A. A. Gončar, The rate of rational approximation and the property of univalence of an analytic function in the neighborhood of an isolated singular point, Mat. Sb. (N.S.) 94(136) (1974), 265–282, 336 (Russian). MR 0352477
  • [Me] Günter Meinardus, Approximation of functions: Theory and numerical methods, Expanded translation of the German edition. Translated by Larry L. Schumaker. Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag New York, Inc., New York, 1967. MR 0217482
  • [Ne] D. J. Newman, Rational approximation to \vert𝑥\vert, Michigan Math. J. 11 (1964), 11–14. MR 0171113
  • [Ri] Theodore J. Rivlin, An introduction to the approximation of functions, Dover Publications, Inc., New York, 1981. Corrected reprint of the 1969 original; Dover Books on Advanced Mathematics. MR 634509
  • [St] G. Shtal′, Best uniform rational approximations of \vert𝑥\vert on [-1,1], Mat. Sb. 183 (1992), no. 8, 85–118 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 76 (1993), no. 2, 461–487. MR 1187250, 10.1070/SM1993v076n02ABEH003422
  • [StTo] Herbert Stahl and Vilmos Totik, General orthogonal polynomials, Encyclopedia of Mathematics and its Applications, vol. 43, Cambridge University Press, Cambridge, 1992. MR 1163828
  • [Tz] Jean Tzimbalario, Rational approximation to 𝑥^{𝛼}, J. Approximation Theory 16 (1976), no. 2, 187–193. MR 0402344
  • [VC1] Richard S. Varga and Amos J. Carpenter, On the Bernstein conjecture in approximation theory, Constr. Approx. 1 (1985), no. 4, 333–348. MR 891763, 10.1007/BF01890040
  • [VC2] -, Some numerical results on best uniform rational approximation of $ {x^{\alpha}}$ on [0, 1], Numer. Algorithms (to appear).
  • [VCR] R. S. Varga, A. Ruttan, and A. Dzh. Karpenter, Numerical results on the best uniform rational approximations of the function \vert𝑥\vert on the interval [-1,1], Mat. Sb. 182 (1991), no. 11, 1523–1541 (Russian); English transl., Math. USSR-Sb. 74 (1993), no. 2, 271–290. MR 1137861
  • [Vj1] N. S. Vjačeslavov, The approximation of the function 𝑥 by rational functions, Mat. Zametki 16 (1974), 163–171 (Russian). MR 0355426
  • [Vj2] N. S. Vjačeslavov, The uniform approximation of 𝑥 by rational functions, Dokl. Akad. Nauk SSSR 220 (1975), 512–515 (Russian). MR 0380214
  • [Vj3] -, On the approximation of $ {x^{\alpha}}$ by rational functions, Izv. Akad. Nauk USSR 44 (1980); English transl. in Math. USSR Izv. 16 (1981), 83-101.

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC: 41A20

Retrieve articles in all journals with MSC: 41A20


Additional Information

DOI: https://doi.org/10.1090/S0273-0979-1993-00351-3
Article copyright: © Copyright 1993 American Mathematical Society