[La propriété reproduisante dans les RKHS généralisée en application aux opérateurs dérivée et intégrale]
In this paper, we consider the reproducing property in Reproducing Kernel Hilbert Spaces (RKHS). We establish a reproducing property for the closure of the class of combinations of composition operators under minimal conditions. This allows to revisit the sufficient conditions for the reproducing property to hold for the derivative operator, as well as for the existence of the mean embedding function. These results provide a framework of application of the representer theorem for regularized learning algorithms that involve data for function values, gradients, or any other operator from the considered class.
Dans cet article, nous considérons la propriété reproduisante dans les espaces de Hilbert à noyaux reproduisants (RKHS). Nous établissons une propriété de reproduction pour l’adhérence de la classe des combinaisons d’opérateurs de composition sous des conditions minimales. Cela nous permet de revisiter les conditions suffisantes pour que la propriété de reproduction soit valable pour l’opérateur dérivé, ainsi que pour l’existence de la fonction mean embedding. Ces résultats donnent un cadre d’application du théorème du représentant pour les algorithmes d’apprentissage régularisés qui impliquent des données sur les valeurs de fonctions, les gradients ou tout autre opérateur de la classe considérée.
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Keywords: RKHS, reproducing property, linear operator, derivative, mean embedding
Mots-clés : RKHS, propriété reproduisante, opérateur linéaire, dérivée, mean embedding
Fatima-Zahrae El-Boukkouri 1 ; Josselin Garnier 2 ; Olivier Roustant 1
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Fatima-Zahrae El-Boukkouri; Josselin Garnier; Olivier Roustant. General reproducing properties in RKHS with application to derivative and integral operators. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 549-561. doi: 10.5802/crmath.847
@article{CRMATH_2026__364_G3_549_0,
author = {Fatima-Zahrae El-Boukkouri and Josselin Garnier and Olivier Roustant},
title = {General reproducing properties in {RKHS} with application to derivative and integral operators},
journal = {Comptes Rendus. Math\'ematique},
pages = {549--561},
year = {2026},
publisher = {Acad\'emie des sciences, Paris},
volume = {364},
doi = {10.5802/crmath.847},
language = {en},
}
TY - JOUR AU - Fatima-Zahrae El-Boukkouri AU - Josselin Garnier AU - Olivier Roustant TI - General reproducing properties in RKHS with application to derivative and integral operators JO - Comptes Rendus. Mathématique PY - 2026 SP - 549 EP - 561 VL - 364 PB - Académie des sciences, Paris DO - 10.5802/crmath.847 LA - en ID - CRMATH_2026__364_G3_549_0 ER -
%0 Journal Article %A Fatima-Zahrae El-Boukkouri %A Josselin Garnier %A Olivier Roustant %T General reproducing properties in RKHS with application to derivative and integral operators %J Comptes Rendus. Mathématique %D 2026 %P 549-561 %V 364 %I Académie des sciences, Paris %R 10.5802/crmath.847 %G en %F CRMATH_2026__364_G3_549_0
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