The distribution of extremal points for Kergin interpolations: real case
Annales de l'Institut Fourier, Tome 48 (1998) no. 1, pp. 205-222.

Nous montrons que les seuls compacts convexes totalement réels de ℂn qui admettent des tableaux extrémaux pour l’interpolation de Kergin sont les ellipses totalement réelles. (Un tableau est dit extrémal pour K lorsqu’il assure la convergence uniforme (sur K) des polynômes d’interpolation vers la fonction interpolée dès que celle-ci est holomorphe au voisinage de K.) Les tableaux extrémaux sur ces ellipses sont caractérisés (en fonction de la distribution des points) et la vitesse de convergence explicitée. Incidemment, nous décrivons le premier exemple (en dimension supérieure) de compact convexe d’intérieur non vide et non circulaire qui admette un tableau extrémal.

We show that a convex totally real compact set in ℂn 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.

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     title = {The distribution of extremal points for {Kergin} interpolations: real case},
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Bloom, Thomas; Calvi, Jean-Paul. The distribution of extremal points for Kergin interpolations: real case. Annales de l'Institut Fourier, Tome 48 (1998) no. 1, pp. 205-222. doi : 10.5802/aif.1615. https://www.numdam.org/articles/10.5802/aif.1615/

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