# Theorems of Korovkin type for adapted spaces

Annales de l'institut Fourier (1973)

- Volume: 23, Issue: 4, page 245-260
- ISSN: 0373-0956

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topBauer, Heinz. "Theorems of Korovkin type for adapted spaces." Annales de l'institut Fourier 23.4 (1973): 245-260. <http://eudml.org/doc/74150>.

@article{Bauer1973,

abstract = {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.},

author = {Bauer, Heinz},

journal = {Annales de l'institut Fourier},

language = {eng},

number = {4},

pages = {245-260},

publisher = {Association des Annales de l'Institut Fourier},

title = {Theorems of Korovkin type for adapted spaces},

url = {http://eudml.org/doc/74150},

volume = {23},

year = {1973},

}

TY - JOUR

AU - Bauer, Heinz

TI - Theorems of Korovkin type for adapted spaces

JO - Annales de l'institut Fourier

PY - 1973

PB - Association des Annales de l'Institut Fourier

VL - 23

IS - 4

SP - 245

EP - 260

AB - 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.

LA - eng

UR - http://eudml.org/doc/74150

ER -

## References

top- [1] V. A. BASKAKOV, Some convergence conditions for linear positive operators, Uspehi Mat. Nauk, 16 (1961), 131-135 (Russian). Zbl0103.28404
- [2] H. BAUER, Šilovscher Rand und Dirichletsches Problem. Ann. Inst. Fourier, 11 (1961), 89-136. Zbl0098.06902MR25 #443
- [3] G. CHOQUET, Lectures on Analysis, Vol. II (Representation Theory), Benjamin, New York-Amsterdam, 1969. Zbl0181.39602
- [4] G. FELBECKER and W. SCHEMPP, A generalization of Bohman-Korov-kin's theorem, Math. Zeitschrift, 122 (1971), 63-70. Zbl0203.13004MR45 #789
- [5] G. FRANCHETTI, Convergenza di operatori in sottospazi dello spacio C(Q), Boll. d. Un. Matem. Ital., Ser. IV, 3 (1970), 668-675. Zbl0199.44403
- [6] M. V. GROSSMAN, Note on a generalized Bohman-Korovkin theorem (to appear in J. of Math. Anal. and Appl.). Zbl0269.41019
- [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. Zbl0050.34005
- [8] P. P. KOROVKIN, Linear operators and approximation theory, Hindustan Publ. Corp., Delhi, India, 1960. Zbl0094.10201
- [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. Zbl0182.16302
- [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). Zbl0117.33003
- [11] Yu A. ŠAŠKIN, Korovkin systems in spaces of continuous functions, Amer. Math. Soc. Transl., Ser. 2, 54 (1966), 125-144. Zbl0178.48601
- [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). Zbl0203.13902
- [14] E. SCHEFFOLD, Über die punktweise Konvergenz von Operatoren in Banachräumen (Manuskript). Zbl0292.41024
- [15] D. E. WULBERT, Convergence of operators and Korovkin's theorem, J. of Appr. Theory, 1 (1968), 381-390. Zbl0167.12904MR38 #3679

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