A Generalised Korovkin Theorem

Pradipta Bandyopadhyay; Ashoke K. Roy

doi:10.68381/jca22044

Abstract

The main result of this paper is a general Korovkin-type approximation theorem for an injective isometry between C(X) and C(Y), where X, Y are compact Hausdorff spaces.

Author information

Contact details are reproduced from the original publication and may be historical.

Pradipta Bandyopadhyay

Statistics Mathematics Division, Indian Statistical Institute, 203 B. T. Road, Kolkata 700 108, India

pradipta@isical.ac.in

Ashoke K. Roy

Dept. of Mathematics, Ramakrishna Mission, Vivekananda University, P. O.: Belur Math, Howrah 711 202, West Bengal – India

ashokeroy2@gmail.com

Keywords

Korovkin-type approximation
Choquet boundary
injective isometry

Mathematics Subject Classification

41A35, 41A36, 46E15

References

F. Altomare, C. Boccacio: On Korovkin-type theorems in spaces of continuous complex-valued functions, Boll. Unione Mat. Ital., VI. Ser., B 1(1) (1982) 75–86.Google Scholar
F. Altomare: Korovkin-type theorems and approximation by positive linear operators, Surv. Approx. Theory 5 (2010) 92–164.Google Scholar
H. Bauer: Approximation and abstract boundaries, Am. Math. Mon. 85 (1978) 632–647.https://doi.org/10.1080/00029890.1978.11994662 Google Scholar
H. Berens, G. G. Lorentz: Geometric theory of Korovkin sets, J. Approx. Theory 15 (1975) 161–189.https://doi.org/10.1016/0021-9045(75)90100-8 Google Scholar
N. Dunford, J. T. Schwartz: Linear Operators, Part 1: General Theory, Interscience, New York (1958).Google Scholar
W. Holsztyński: Continuous mappings induced by isometries of spaces of continuous functions, Stud. Math. 26 (1966) 133–136.https://doi.org/10.4064/sm-26-2-133-136 Google Scholar
C. A. Micchelli: Convergence of positive linear operators on C(X), J. Approx. Theory 13 (1975) 305–315.https://doi.org/10.1016/0021-9045(75)90040-4 Google Scholar
R. R. Phelps: Lectures on Choquet’s Theorem, 2nd Ed., Lecture Notes in Math. 1757, Springer, Berlin (2001).https://doi.org/10.1007/b76887 Google Scholar
R. R. Phelps: The Choquet representation in the complex case, Bull. Amer. Math. Soc. 83 (1977) 299–312.https://doi.org/10.1090/S0002-9904-1977-14243-2 Google Scholar
Sin-ei Takahasi: BKW-operators on function spaces, Rend. Circ. Mat. Palermo (2) Suppl. 33 (1993) 479–488.Google Scholar
Sin-ei Takahasi: (T, E)-Korovkin closures in normed spaces and BKW-operators, J. Approx. Theory 82(3) (1995) 340–351.https://doi.org/10.1006/jath.1995.1083 Google Scholar
D. E. Wulbert: Convergence of operators and Korovkin’s theorem, J. Approx. Theory 1 (1968) 381–390.https://doi.org/10.1016/0021-9045(68)90016-6 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.

[1] F. Altomare, C. Boccacio: On Korovkin-type theorems in spaces of continuous complex-valued functions, Boll. Unione Mat. Ital., VI. Ser., B 1(1) (1982) 75–86.Google Scholar

[2] F. Altomare: Korovkin-type theorems and approximation by positive linear operators, Surv. Approx. Theory 5 (2010) 92–164.Google Scholar

[3] H. Bauer: Approximation and abstract boundaries, Am. Math. Mon. 85 (1978) 632–647.https://doi.org/10.1080/00029890.1978.11994662 Google Scholar

[4] H. Berens, G. G. Lorentz: Geometric theory of Korovkin sets, J. Approx. Theory 15 (1975) 161–189.https://doi.org/10.1016/0021-9045(75)90100-8 Google Scholar

[5] N. Dunford, J. T. Schwartz: Linear Operators, Part 1: General Theory, Interscience, New York (1958).Google Scholar

[6] W. Holsztyński: Continuous mappings induced by isometries of spaces of continuous functions, Stud. Math. 26 (1966) 133–136.https://doi.org/10.4064/sm-26-2-133-136 Google Scholar

[7] C. A. Micchelli: Convergence of positive linear operators on C(X), J. Approx. Theory 13 (1975) 305–315.https://doi.org/10.1016/0021-9045(75)90040-4 Google Scholar

[8] R. R. Phelps: Lectures on Choquet’s Theorem, 2nd Ed., Lecture Notes in Math. 1757, Springer, Berlin (2001).https://doi.org/10.1007/b76887 Google Scholar

[9] R. R. Phelps: The Choquet representation in the complex case, Bull. Amer. Math. Soc. 83 (1977) 299–312.https://doi.org/10.1090/S0002-9904-1977-14243-2 Google Scholar

[10] Sin-ei Takahasi: BKW-operators on function spaces, Rend. Circ. Mat. Palermo (2) Suppl. 33 (1993) 479–488.Google Scholar

[11] Sin-ei Takahasi: (T, E)-Korovkin closures in normed spaces and BKW-operators, J. Approx. Theory 82(3) (1995) 340–351.https://doi.org/10.1006/jath.1995.1083 Google Scholar

[12] D. E. Wulbert: Convergence of operators and Korovkin’s theorem, J. Approx. Theory 1 (1968) 381–390.https://doi.org/10.1016/0021-9045(68)90016-6 Google Scholar