Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-04T19:21:14.579Z Has data issue: false hasContentIssue false
  • English
  • Français

On the approximation of certain functions by interpolating polynomials

Published online by Cambridge University Press:  17 April 2009

Michael Revers
Michael Revers
Affiliation:
Department of Mathematics, University of Salzburg, Hellbrunnerstrasse 34, A–5020 Salzburg, Austria e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the present paper we consider the approximation of |x|α on [−1, 1] by interpolating polynomials and we establish upper bounds for the approximation error. It turns out that these bounds, apart from the constants, are of the best possible order. We compare our results with estimates for the best approximation, established by Bernstein.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 58 , Issue 3 , December 1998 , pp. 505 - 512
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Bell, R.A. and Shah, S.M., ‘Oscillating polynomials and approximation to fractional powers of x’, J. Approx. Theory 1 (1968), 269–274.CrossRefGoogle Scholar
[2]Bernstein, S., ‘Sur la meilleure approximation de |x| par des polynômes de degrés donnés’, Acta Math. 37 (1913), 1–57.CrossRefGoogle Scholar
[3]Bernstein, S., ‘Sur la meilleure approximation de |x|p par des polynômes de degrés trés élevés’, Bull. Acad. Sci. USSR Sér. Math. 2 (1938), 181–190.Google Scholar
[4]Bernstein, S., ‘On the best approximation of |x|p by means of polynomials of extremely high degree’, (in Russian), Izv. Akad. Nauk. SSSR 2 (1938), 169–180.Google Scholar
[5]Brutman, L. and Passow, E., ‘Rational interpolation to |x| at the Chebyshev nodes’, Bull. Austral. Math. Soc. 56 (1997), 81–86.CrossRefGoogle Scholar
[6]Byrne, G.J., Mills, T.M. and Smith, S.J., ‘On Lagrange interpolation with equidistant nodes’, Bull. Austral. Math. Soc. 42 (1990), 81–89.CrossRefGoogle Scholar
[7]Elosser, P.D., ‘Approximation of powers of x by polynomials’, J. Approx. Theory 23 (1978), 163–174.CrossRefGoogle Scholar
[8]Elosser, P.D., ‘Chebychev coefficients in approximation of powers of x’ J. Approx. Theory 63 (1990), 293–302.CrossRefGoogle Scholar
[9]Newman, D.J., ‘Rational approximation to |x|’, Michigan Math. J. 11 (1964), 11–14.CrossRefGoogle Scholar
[10]Stahl, H., ‘Best uniform rational approximation of x α on [0, 1]’, Bull. Amer. Math. Soc. 28 (1993), 116–122.CrossRefGoogle Scholar
[11]Varga, R.S. and Carpenter, A.J., ‘On the Bernstein Conjecture in approximation theory’, Const. Approx. 1 (1985), 333–348.CrossRefGoogle Scholar