We show that a convex totally real compact set in 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 when the corresponding sequence of Kergin interpolation polynomials converges uniformly (on ) to the interpolated function as soon as it is holomorphic on a neighborhood of .). 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.
Nous montrons que les seuls compacts convexes totalement réels de qui admettent des tableaux extrémaux pour l’interpolation de Kergin sont les ellipses totalement réelles. (Un tableau est dit extrémal pour lorsqu’il assure la convergence uniforme (sur ) des polynômes d’interpolation vers la fonction interpolée dès que celle-ci est holomorphe au voisinage de .) 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.
@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}, pages = {205--222}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {48}, number = {1}, year = {1998}, doi = {10.5802/aif.1615}, zbl = {0915.41001}, mrnumber = {99c:32015}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1615/} }
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 SP - 205 EP - 222 VL - 48 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1615/ DO - 10.5802/aif.1615 LA - en ID - AIF_1998__48_1_205_0 ER -
%0 Journal Article %A Bloom, Thomas %A Calvi, Jean-Paul %T The distribution of extremal points for Kergin interpolations: real case %J Annales de l'Institut Fourier %D 1998 %P 205-222 %V 48 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1615/ %R 10.5802/aif.1615 %G en %F 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, Volume 48 (1998) no. 1, pp. 205-222. doi : 10.5802/aif.1615. https://aif.centre-mersenne.org/articles/10.5802/aif.1615/
[1] Complex Kergin interpolation, J. Approx. Theory, 64 (1991), 214-225. | MR | Zbl
and ,[2] Complex Kergin interpolation and the Fantappiè transform, Math. Z., 208 (1991), 257-271. | MR | Zbl
and ,[3] Complex convexity and analytic functionals I, preprint, University of Iceland, 1995.
, and ,[4] Kergin interpolants of Holomorphic Functions, Constr. Approx., 13 (1997), 569-583. | MR | Zbl
and ,[5] Theorie der Konvexen Körper, Chelsea, New York, 1971. | Zbl
and ,[6] Kergin interpolants at the roots of unity approximate C2 functions, J. Analyse Math., (to appear). | Zbl
and ,[7] Lectures on complex approximation, Birkhauser, Boston, 1987. | MR | Zbl
,[8] Pluripotential Theory, Oxford University Press, Oxford, 1991. | MR | Zbl
,[9] Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972. | MR | Zbl
,[10] An extremal plurisubharmonic function associated to a convex pluricomplex Green function with pole at infinity, J. reine angew. Math., 471 (1996), 139-163. | MR | Zbl
,[11] Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995. | MR | Zbl
,[12] Extremal plurisubharmonic functions on ℂN, Ann. Pol. Math., XXXIX (1981), 175-211. | MR | Zbl
,[13] General orthogonal polynomials, Cambridge University Press, Cambridge, 1992. | MR | Zbl
and ,[14] Representations of functionals via summability methods. I, Acta. Sci. Math., 48 (1985), 483-498. | MR | Zbl
,[15] Interpolation and Approximation by Rational Functions in the Complex Domain (5th edition), A.M.S., Providence, 1969.
,Cited by Sources: