Abstract
For a function f(z) analytic on |z| < ρ, ρ > 1, we consider two schemes of rational interpolants which have poles equally spaced on the circle |z|=σ, σ > 1. The first scheme interpolates f(z) in the roots of unity, while the second consists of best L2-approximants to f(z) on the unit circle. We obtain precise regions of equiconvergence for the two schemes of rational functions, thus extending a well-known result of J. L. Walsh.
Research supported in part by the National Science Foundation.
Research supported in part by the National Research Council of Canada.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. S. Cavaretta Jr., A. Sharma and R. S. Varga. Interpolation in the roots of unity: An extension of a theorem of J. L. Walsh. Resultate der Math. 1981, vol. 3, 155–191.
G. López Lagomasino and René Piedra de la Torre, Sbore un teorema de sobreconvergencia de J. L. Walsh. Revista Ciencias Matemáticas. 1983, vol. IV, No. 3, 67–78.
E. B. Staff, A. Sharma and R. S. Varga, An extension to rational functions of a theorem of J. L. Walsh on differences of interpolating polynomials, R.A.I.R.O. Analyse numérique, 1981, vol. 15, 371–390.
R. S. Varga. Topics in Polynomial and Rational Interpolation and Approximation. Séminaire de Math. Supérieures, Les Presses de L'Univ. de Montréal, Montréal, 1982, 69–93.
J. L. Walsh. Interpolation and Approximation by Rational Functions in the Complex Domain. A.M.S. Colloq. Publ. Vol XX, Providence, R.I., 5th ed. 1969.
A. Zygmund, Trigonometric Series, Vol. II, University Press, Cambridge, 2nd ed. 1959.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1984 Springer-Verlag
About this paper
Cite this paper
Saff, E.B., Sharma, A. (1984). On equiconvergence of certain sequences of rational interpolants. In: Graves-Morris, P.R., Saff, E.B., Varga, R.S. (eds) Rational Approximation and Interpolation. Lecture Notes in Mathematics, vol 1105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072417
Download citation
DOI: https://doi.org/10.1007/BFb0072417
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-13899-0
Online ISBN: 978-3-540-39113-5
eBook Packages: Springer Book Archive