Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-29T12:57:11.500Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic distribution of the zeros of certain Lagrange interpolants

Published online by Cambridge University Press:  20 January 2009

M. A. Bokhari  and
M. Iqbal
M. A. Bokhari
Affiliation:
Department of Mathematical SciencesKing Fahd University of Petroleum and MineralsDhahran 31261, Saudi Arabia E-Mail: FACL049@SAUPM00.BITNET.
M. Iqbal
Affiliation:
Department of Mathematical SciencesKing Fahd University of Petroleum and MineralsDhahran 31261, Saudi Arabia E-Mail: FACL049@SAUPM00.BITNET.
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.

We consider rational functions of the form fm(z) = zm/(z – p) which are analytic in |z|<p, p>1, and establish that the asymptotic distribution of the zeros of their Taylor sections and Lagrange interpolants at uniformly distributed nodes is similar. This notion is also illustrated computationally. We conjecture that a similar result can be expected for any function analytic in |z| < p.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 38 , Issue 1 , February 1995 , pp. 99 - 106
Copyright
Copyright © Edinburgh Mathematical Society 1995

References

REFERENCES

1.Bokhari, M. A., On certain sequences of least squares approximants, Bull. Austral. Math. Soc. 38 (1988), 415–422.CrossRefGoogle Scholar
2.Bokhari, M. A., Equiconvergence of some sequences of complex interpolating rational functions, J. Approx. Th. 55 (1988), 205–219.CrossRefGoogle Scholar
3.Bokhari, M. A., Converse results in the theory of equiconvergence of interpolating rational functions, Rocky Mount. J. Math. 19 (1989), 73–81.CrossRefGoogle Scholar
4.Cavaretta, A. S., Sharma, A. and Varga, R. S., Interpolation in the roots of unity: an extension of a theorem of J.L. Walsh, Resultate Math. 3 (1981), 155–191.CrossRefGoogle Scholar
5.Edrei, A., Saff, E. B. and Varga, R. S., Zeros of Sections of Power Series (Lecture Notes in Mathematics 1002, Springer-Berlag, Berlin, 1983).CrossRefGoogle Scholar
6.Jentzsch, R., Untersuchungen zur Theorie der Folgen analytischer Functionen, Acta Math. 41 (1917), 219–251.CrossRefGoogle Scholar
7.Marden, M., Geometry of Polynomials (Mathematical Surveys No. 3, Amer. Math. Soc., Providence, RI, 1966).Google Scholar
8.Rosenbloom, P. C., Distribution of Zeros of Polynomials, in Lectures on Functions of a Complex Variable (Kaplan, W., ed., University of Michigan Press, Ann Arbor, 1955), 265–285.Google Scholar
9.Szegö, G., Über die Nullstellen von Polynomen die in einem kreise gleichmässig kon-vergieren, Sitzungsber. Berl. Math. Ges. 21 (1922), 59–64.Google Scholar
10.Varga, R. S., Topics in polynomial and rational interpolation and approximation chapter IV. Seminaire de Math. Superieures, Monreal, 1982).Google Scholar
11.Varga, R. S., Scientific Computation on Mathematical Problems and Conjectures (CBMS-NSF Regional Conference Series in Applied Math., SIAM, Philadelphia, 1990).CrossRefGoogle Scholar
12.Walsh, J. L., Interpolation and approximation by rational functions in the complex domain (A.M.S. Colloq. Publications, Vol. XX, Providence, R.I., 5th ed., 1969).Google Scholar