Theorems of Korovkin type for adapted spaces

Heinz Bauer

Annales de l'institut Fourier (1973)

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

Abstract

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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.

How to cite

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Bauer, 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

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  1. [1] V. A. BASKAKOV, Some convergence conditions for linear positive operators, Uspehi Mat. Nauk, 16 (1961), 131-135 (Russian). Zbl0103.28404
  2. [2] H. BAUER, Šilovscher Rand und Dirichletsches Problem. Ann. Inst. Fourier, 11 (1961), 89-136. Zbl0098.06902MR25 #443
  3. [3] G. CHOQUET, Lectures on Analysis, Vol. II (Representation Theory), Benjamin, New York-Amsterdam, 1969. Zbl0181.39602
  4. [4] G. FELBECKER and W. SCHEMPP, A generalization of Bohman-Korov-kin's theorem, Math. Zeitschrift, 122 (1971), 63-70. Zbl0203.13004MR45 #789
  5. [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. [6] M. V. GROSSMAN, Note on a generalized Bohman-Korovkin theorem (to appear in J. of Math. Anal. and Appl.). Zbl0269.41019
  7. [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. [8] P. P. KOROVKIN, Linear operators and approximation theory, Hindustan Publ. Corp., Delhi, India, 1960. Zbl0094.10201
  9. [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. [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. [11] Yu A. ŠAŠKIN, Korovkin systems in spaces of continuous functions, Amer. Math. Soc. Transl., Ser. 2, 54 (1966), 125-144. Zbl0178.48601
  12. [12] Yu A. ŠAŠKIN, The Milman-Choquet boundary and approximation theory, Functional Anal. Appl., 1 (1967), 170-171. 
  13. [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. [14] E. SCHEFFOLD, Über die punktweise Konvergenz von Operatoren in Banachräumen (Manuskript). Zbl0292.41024
  15. [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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