The distribution of extremal points for Kergin interpolations : real case

Thomas Bloom; Jean-Paul Calvi

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 1, page 205-222
  • ISSN: 0373-0956

Abstract

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We show that a convex totally real compact set in ℂ n admits an extremal array for Kergin interpolation if and only if it is a totally real ellipse. (An array is said to be extremal for K when the corresponding sequence of Kergin interpolation polynomials converges uniformly (on K ) to the interpolated function as soon as it is holomorphic on a neighborhood of K .). Extremal arrays on these ellipses are characterized in terms of the distribution of the points and the rate of convergence is investigated. In passing, we construct the first (higher dimensional) example of a compact convex set of non void interior that admits an extremal array without being circular.

How to cite

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Bloom, Thomas, and Calvi, Jean-Paul. "The distribution of extremal points for Kergin interpolations : real case." Annales de l'institut Fourier 48.1 (1998): 205-222. <http://eudml.org/doc/75276>.

@article{Bloom1998,
abstract = {We show that a convex totally real compact set in $\Bbb C^n$ admits an extremal array for Kergin interpolation if and only if it is a totally real ellipse. (An array is said to be extremal for $K$ when the corresponding sequence of Kergin interpolation polynomials converges uniformly (on $K$) to the interpolated function as soon as it is holomorphic on a neighborhood of $K$.). Extremal arrays on these ellipses are characterized in terms of the distribution of the points and the rate of convergence is investigated. In passing, we construct the first (higher dimensional) example of a compact convex set of non void interior that admits an extremal array without being circular.},
author = {Bloom, Thomas, Calvi, Jean-Paul},
journal = {Annales de l'institut Fourier},
keywords = {Kergin interpolation; polynomial approximation of holomorphic functions; logarithmic potentials; totally real sets; convergence criterion; rate of convergence},
language = {eng},
number = {1},
pages = {205-222},
publisher = {Association des Annales de l'Institut Fourier},
title = {The distribution of extremal points for Kergin interpolations : real case},
url = {http://eudml.org/doc/75276},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Bloom, Thomas
AU - Calvi, Jean-Paul
TI - The distribution of extremal points for Kergin interpolations : real case
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 1
SP - 205
EP - 222
AB - We show that a convex totally real compact set in $\Bbb C^n$ admits an extremal array for Kergin interpolation if and only if it is a totally real ellipse. (An array is said to be extremal for $K$ when the corresponding sequence of Kergin interpolation polynomials converges uniformly (on $K$) to the interpolated function as soon as it is holomorphic on a neighborhood of $K$.). Extremal arrays on these ellipses are characterized in terms of the distribution of the points and the rate of convergence is investigated. In passing, we construct the first (higher dimensional) example of a compact convex set of non void interior that admits an extremal array without being circular.
LA - eng
KW - Kergin interpolation; polynomial approximation of holomorphic functions; logarithmic potentials; totally real sets; convergence criterion; rate of convergence
UR - http://eudml.org/doc/75276
ER -

References

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  11. [11] T. RANSFORD, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995. Zbl0828.31001MR96e:31001
  12. [12] J. SICIAK, Extremal plurisubharmonic functions on ℂN, Ann. Pol. Math., XXXIX (1981), 175-211. Zbl0477.32018MR83e:32018
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