The distribution of extremal points for Kergin interpolations: real case
Annales de l'Institut Fourier, Tome 48 (1998) no. 1, p. 205-222
Nous montrons que les seuls compacts convexes totalement réels de n 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 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.
@article{AIF_1998__48_1_205_0,
     author = {Bloom, Thomas and Calvi, Jean-Paul},
     title = {The distribution of extremal points for Kergin interpolations: real case},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {48},
     number = {1},
     year = {1998},
     pages = {205-222},
     doi = {10.5802/aif.1615},
     zbl = {0915.41001},
     mrnumber = {99c:32015},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1998__48_1_205_0}
}
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. http://www.numdam.org/item/AIF_1998__48_1_205_0/

[1] M. Andersson and M. Passare, Complex Kergin interpolation, J. Approx. Theory, 64 (1991), 214-225. | MR 92b:41003 | Zbl 0737.32007

[2] M. Andersson and M. Passare, Complex Kergin interpolation and the Fantappiè transform, Math. Z., 208 (1991), 257-271. | MR 92j:32046 | Zbl 0725.32011

[3] M. Andersson, M. Passare and R. Sigurdsson, Complex convexity and analytic functionals I, preprint, University of Iceland, 1995.

[4] T. Bloom and J.-P. Calvi, Kergin interpolants of Holomorphic Functions, Constr. Approx., 13 (1997), 569-583. | MR 98h:32021 | Zbl 0904.32011

[5] T. Bonnensen and W. Fenchel, Theorie der Konvexen Körper, Chelsea, New York, 1971. | Zbl 0906.52001

[6] L. Bos and J.-P. Calvi, Kergin interpolants at the roots of unity approximate C2 functions, J. Analyse Math., (to appear). | Zbl 0901.41002

[7] D. Gaier, Lectures on complex approximation, Birkhauser, Boston, 1987. | MR 88i:30059b | Zbl 0612.30003

[8] M. Klimek, Pluripotential Theory, Oxford University Press, Oxford, 1991. | MR 93h:32021 | Zbl 0742.31001

[9] N.S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972. | MR 50 #2520 | Zbl 0253.31001

[10] S. Momm, An extremal plurisubharmonic function associated to a convex pluricomplex Green function with pole at infinity, J. reine angew. Math., 471 (1996), 139-163. | MR 97b:32013 | Zbl 0848.31008

[11] T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995. | MR 96e:31001 | Zbl 0828.31001

[12] J. Siciak, Extremal plurisubharmonic functions on ℂN, Ann. Pol. Math., XXXIX (1981), 175-211. | MR 83e:32018 | Zbl 0477.32018

[13] H. Stahl and V. Totik, General orthogonal polynomials, Cambridge University Press, Cambridge, 1992. | MR 93d:42029 | Zbl 0791.33009

[14] V. Totik, Representations of functionals via summability methods. I, Acta. Sci. Math., 48 (1985), 483-498. | MR 87f:46046 | Zbl 0594.46021

[15] J.L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain (5th edition), A.M.S., Providence, 1969.