Skip to main content
×
×
Home

Ray Sequences of Best Rational Approximants For |x|α

  • E. B. Saff (a1) and H. Stahl (a2)
Abstract

The convergence behavior of best uniform rational approximations with numerator degree m and denominator degree n to the function |x|α, α > 0, on [-1, 1] is investigated. It is assumed that the indices (m, n) progress along a ray sequence in the lower triangle of the Walsh table, i.e. the sequence of indices {(m, n)} satisfies

In addition to the convergence behavior, the asymptotic distribution of poles and zeros of the approximants and the distribution of the extreme points of the error function on [-1, 1] will be studied. The results will be compared with those for paradiagonal sequences (m = n + 2[α/2]) and for sequences of best polynomial approximants.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Ray Sequences of Best Rational Approximants For |x|α
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Ray Sequences of Best Rational Approximants For |x|α
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Ray Sequences of Best Rational Approximants For |x|α
      Available formats
      ×
Copyright
References
Hide All
[An] Anderson, J.-E., Rational approximation to functions like xα in integral norms, Anal. Math. 14(1988), 1125.
[Be1] Bernstein, S., About the best approximation of |x|p by means of polynomials of very high degree, (Russian), Collected Works II(1938), 262272.
[Be2] Bernstein, S., Sur meilleure approximation de |x. par des polynômes degrés donnés, Acta Math. 37(1913), 157.
[BIS] Blatt, H.-P., Iserles, A. and Saff, E.B., Remarks on the behavior of zeros and poles of best approximating polynomials and rational functions. In: Algorithms for Approximation, (eds. Mason, J.C. and Cox, M.G.), Inst. Math. Appl. Conf. Ser. New Ser. 10, Claredon Press, Oxford, 1987. 437445.
[BS] Blatt, H.-P. and Saff, E.B., Behavior of zeros of polynomials of near best approximation, J.Approx. Theory 46(1986), 323344.
[Bu1] Bulanow, A.P., Asymptotics for the least derivation of |x. from rational functions, Mat. Sb. (118) 76(1968), 288303. English transl. in Math. USSR-Sb. 5(1968), 275290.
[Bu2] Bulanow, A.P., The approximation of x1/3 by rational functions, (Russian), Vests¯ı Akad. Navuk Belarus¯ı Ser. F¯ız. Mat. Navuk 2(1968), 4756.
[FrSz] Freud, G. and Szabados, J., Rational approximation to xα, ActaMath. Acad. Sci. Hungar. 18(1967), 393.
[Ga] Ganelius, T., Rational approximation to x α on [0, 1], Anal. Math. 5(1979), 1933.
[Ge] Gelfond, A.O., Differenzenrechnung. Deutscher Verlag der Wissenschaften, Berlin, 1958.
[Go1] Gonchar, A.A., On the speed of rational approximation of continuous functions with characteristic singularities, Mat. Sb. (115) 73(1967), 630638. English transl. in Math. USSR-Sb. 2(1967).
[Go2] Gonchar, A.A., Rational approximation of the function xα. (Russian), In: Constructive Theory of Functions, Proc. Internat. Conf, Varna, 1970. Izdat. Bolgar. Akad. Nauk, Sofia, 1972. 5153.
[Go3] Gonchar, A.A., 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. (136) 94(1974), 265282. English transl. in Math. USSR-Sb. 23(1974).
[Ka] Kadec, M.I., On the distribution of points of maximum deviation in the approximation of continuous functions by polynomials, Amer. Math. Soc. Transl. (2) 26(1963), 231234.
[KaSt] Karlin, S. and Studden, W.J., Tchebycheff Systems: With Applications in Analysis and Statistics. Interscience Publishers, New York, 1966.
[La] Landkof, N.S., Foundations of Modern Potential Theory. GrundlehrenMath.Wiss. 190, Springer-Verlag, New York, 1972.
[Me] Meinardus, G., Approximation of Functions: Theory and NumericalMethods. Springer-Verlag,NewYork, 1967.
[Ne] Newman, D.J., Rational approximation to |x|, Michigan Math. J. 11(1964), 1114.
[Ri] Rivlin, T.J., An Introduction to the Approximation of Functions. Blaisdell Publ. Co., Waltham, Massachusetts, 1969.
[Sa] Saff, E.B., A principle of contamination in best polynomial approximation. In: Approximation and Optimization, Lecture Notes in Math. 1354, (eds. Gomez, Guerra, Jimeniz and Lopez), Springer-Verlag, Berlin, 1988. 7997.
[SaSt1] Saff, E.B. and Stahl, H., Sequences in the Walsh table for xα. In: Constructive Theory of Functions, (eds. Ivanov, K., Petrushev, P. and Bl. Sendov), Bulgarian Academy of Science, Sofia, 1992. 249259.
[SaSt2] Saff, E.B., Asymptotic distribution of poles and zeros of best rational approximants for |x|α, Proc. of the Semester of Funct. Theory at the Internat. Banach Center, Warsaw, 1992. to appear.
[St1] Stahl, H., Best uniform rational approximation of |x| on [-1, 1], Mat. Sb. (8) 183, 85118.
[St2] Stahl, H., Best uniform rational approximation of xα on [0, 1], Bull. Amer.Math. Soc. 28(1993), 116122.
[StTo] Stahl, H. and Totik, V., General Orthogonal Polynomials. Encyclopedia Math. Appl. 43, Cambridge University Press, 1992.
[Ts] Tsuji, M., Potential Theory in Modern Function Theory. Maruzen, Tokyo, 1959.
[Tz] Tzimbalario, J., Rational approximation to x α, J. Approx. Theory 16(1976), 187193.
[VC1] Varga, R.S. and Carpenter, A.J., On the Bernstein conjecture in approximation theory, Constr. Approx. 1(1985), 333348. Russian transl. in Mat. Sb. (171) 129(1986), 535548.
[VC2] Varga, R.S., Some numerical results on best uniform rational approximation of xα on [0, 1], Numer. Algorithms, to appear.
[VC3] Varga, R.S., Some numerical results on best uniform polynomial approximation of xα on [0, 1]. In: Methods of Approximation Theory in Complex Analysis and Mathematical Physics, (eds. Gonchar, A.A. and Saff, E.B.), Moskow, “Nauka”, 1992. 192222.
[VRC] Varga, R.S., Ruttan, A. and Carpenter, A.J., Numerical results on best uniform rational approximation of |x. on [-1, 1], Mat. Sb. (11) 182(1991), 15231541.
[Vj1] Vjacheslavov, N.S., On the approximation of xα by rational functions, Izv. Akad. Nauk-USSR 44(1980); English transl. in Math. USSR-Izv. 16(1981), 83101.
[Vj2] Vjacheslavov, N.S., On the uniform approximation of |x. by rational functions, Dokl. Akad. Nauk SSSR 220(1975), 512515. English transl. in Soviet Math. Dokl. 16(1975), 100104.
[Vj3] Vjacheslavov, N.S., The approximation of|x. by rational functions, (Russian), Mat. Zametki 16(1974), 163171.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed