Theorems of Korovkin type for adapted spaces
Annales de l'Institut Fourier, Volume 23 (1973) no. 4, pp. 245-260.

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.

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.

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Bauer, Heinz. Theorems of Korovkin type for adapted spaces. Annales de l'Institut Fourier, Volume 23 (1973) no. 4, pp. 245-260. doi : 10.5802/aif.490. https://aif.centre-mersenne.org/articles/10.5802/aif.490/

[1] V. A. Baskakov, Some convergence conditions for linear positive operators, Uspehi Mat. Nauk, 16 (1961), 131-135 (Russian). | Zbl: 0103.28404

[2] H. Bauer, Šilovscher Rand und Dirichletsches Problem. Ann. Inst. Fourier, 11 (1961), 89-136. | Numdam | MR: 25 #443 | Zbl: 0098.06902

[3] G. Choquet, Lectures on Analysis, Vol. II (Representation Theory), Benjamin, New York-Amsterdam, 1969. | Zbl: 0181.39602

[4] G. Felbecker and W. Schempp, A generalization of Bohman-Korov-kin's theorem, Math. Zeitschrift, 122 (1971), 63-70. | MR: 45 #789 | Zbl: 0203.13004

[5] G. Franchetti, Convergenza di operatori in sottospazi dello spacio C(Q), Boll. d. Un. Matem. Ital., Ser. IV, 3 (1970), 668-675. | Zbl: 0199.44403

[6] M. V. Grossman, Note on a generalized Bohman-Korovkin theorem (to appear in J. of Math. Anal. and Appl.). | Zbl: 0269.41019

[7] P. P. Korovkin, 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] P. P. Korovkin, Linear operators and approximation theory, Hindustan Publ. Corp., Delhi, India, 1960.

[9] G. Mokobodzki et D. Sibony, 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] Yu A. Šaškin, 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] Yu A. Šaškin, Korovkin systems in spaces of continuous functions, Amer. Math. Soc. Transl., Ser. 2, 54 (1966), 125-144. | Zbl: 0178.48601

[12] Yu A. Šaškin, The Milman-Choquet boundary and approximation theory, Functional Anal. Appl., 1 (1967), 170-171.

[13] Yu A. Šaškin, On the convergence of linear operators. Proc. Intern. Conference on Constructive Function Theory, Varna (1970), 119-125 (Russian). | Zbl: 0203.13902

[14] E. Scheffold, Über die punktweise Konvergenz von Operatoren in Banachräumen (Manuskript). | Zbl: 0292.41024

[15] D. E. Wulbert, Convergence of operators and Korovkin's theorem, J. of Appr. Theory, 1 (1968), 381-390. | MR: 38 #3679 | Zbl: 0167.12904

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