[Interpolation with constraints on the finite sets of the disc]
Given a finite set σ of the unit disc and a holomorphic function f in which belongs to a class X, we are looking for a function g in another class Y (smaller than X) which minimizes the norm among all functions g such that . For , and for the corresponding interpolation constant , we show that where , and where stands for the norm of the evaluation functional on the space X. The upper bound is sharp over sets σ with given n and r.
Étant donné un ensemble fini σ du disque unité et une fonction f holomorphe dans appartenant à une certaine classe X, on cherche g dans une autre classe Y (plus petite que X) qui minimise la norme de g dans Y parmi toutes les fonctions g satisfaisant la condition . On montre que dans le cas , la constante d'interpolation correspondante est majorée par où , et est la norme de la fonctionnelle d'évaluation , sur l'espace X. La majoration est exacte sur l'ensemble des σ avec n et r donné.
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Rachid Zarouf 1
@article{CRMATH_2009__347_13-14_785_0, author = {Rachid Zarouf}, title = {Interpolation avec contraintes sur des ensembles finis du disque}, journal = {Comptes Rendus. Math\'ematique}, pages = {785--790}, publisher = {Elsevier}, volume = {347}, number = {13-14}, year = {2009}, doi = {10.1016/j.crma.2009.04.014}, language = {fr}, }
Rachid Zarouf. Interpolation avec contraintes sur des ensembles finis du disque. Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 785-790. doi : 10.1016/j.crma.2009.04.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.014/
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