On montre que les méthodes développées dans un travail antérieur de l’auteur sur le problème de Dirichlet pour la frontière de Silov [Annales Inst. Fourier, 11 (1961)] permettent de retrouver d’une manière nouvelle et naturelle les résultats les plus importants sur la convergence d’opérateurs positifs linéaires dans les espaces de fonctions continues sur un espace compact.
La notion d’un espace adapté de fonctions continues, introduite par Choquet, en liaison avec des résultats de Mokobodzki-Sibony permettent une extension au cas des espaces localement compacts. En particulier, le problème d’une caractérisation de la fermeture de Korovkin d’un espace adapté est résolu.
It is shown that the methods developed in an earlier paper of the author about a Dirichlet problem for the Silov boundary [Annales Inst. Fourier, 11 (1961)] lead in a new and natural way to the most important results about the convergence of positive linear operators on spaces of continuous functions defined on a compact space. Choquet’s notion of an adapted space of continuous functions in connection with results of Mokobodzki-Sibony opens the possibility of extending these results to the case of locally compact spaces. In particular, the so-called Korovkin closure of an adapted space is characterized.
@article{AIF_1973__23_4_245_0,
author = {Bauer, Heinz},
title = {Theorems of Korovkin type for adapted spaces},
journal = {Annales de l'Institut Fourier},
volume = {23},
year = {1973},
pages = {245-260},
doi = {10.5802/aif.490},
mrnumber = {50 \#10643},
zbl = {0262.31005},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_1973__23_4_245_0}
}
Bauer, Heinz. Theorems of Korovkin type for adapted spaces. Annales de l'Institut Fourier, Tome 23 (1973) pp. 245-260. doi : 10.5802/aif.490. http://gdmltest.u-ga.fr/item/AIF_1973__23_4_245_0/
[1] , Some convergence conditions for linear positive operators, Uspehi Mat. Nauk, 16 (1961), 131-135 (Russian). | Zbl 0103.28404
[2] , Šilovscher Rand und Dirichletsches Problem. Ann. Inst. Fourier, 11 (1961), 89-136. | Numdam | MR 25 #443 | Zbl 0098.06902
[3] , Lectures on Analysis, Vol. II (Representation Theory), Benjamin, New York-Amsterdam, 1969. | Zbl 0181.39602
[4] and , A generalization of Bohman-Korov-kin's theorem, Math. Zeitschrift, 122 (1971), 63-70. | MR 45 #789 | Zbl 0203.13004
[5] , Convergenza di operatori in sottospazi dello spacio C(Q), Boll. d. Un. Matem. Ital., Ser. IV, 3 (1970), 668-675. | Zbl 0199.44403
[6] , Note on a generalized Bohman-Korovkin theorem (to appear in J. of Math. Anal. and Appl.). | Zbl 0269.41019
[7] , On convergence of linear positive operators in the space of continuous functions, Doklady Akad. Nauk SSSR (N.S.), 90 (1953), 961-964. | Zbl 0050.34005
[8] , Linear operators and approximation theory, Hindustan Publ. Corp., Delhi, India, 1960.
[9] et , Cônes adaptés de fonctions continues et théorie du potentiel. Séminaire Choquet, Initiation à l'Analyse, 6e année (1966/1967), Fasc. 1, 35 p., Institut H.-Poincaré, Paris, 1968. | Numdam | Zbl 0182.16302
[10] , On the convergence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR, 131 (1960), 525-527 (Russian). | Zbl 0117.33003
[11] , Korovkin systems in spaces of continuous functions, Amer. Math. Soc. Transl., Ser. 2, 54 (1966), 125-144. | Zbl 0178.48601
[12] , The Milman-Choquet boundary and approximation theory, Functional Anal. Appl., 1 (1967), 170-171.
[13] , On the convergence of linear operators. Proc. Intern. Conference on Constructive Function Theory, Varna (1970), 119-125 (Russian). | Zbl 0203.13902
[14] , Über die punktweise Konvergenz von Operatoren in Banachräumen (Manuskript). | Zbl 0292.41024
[15] , Convergence of operators and Korovkin's theorem, J. of Appr. Theory, 1 (1968), 381-390. | MR 38 #3679 | Zbl 0167.12904