Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Sequential type Korovkin theorem on $L^\infty $
for $\textbf {QC}$-test functions


Author: Keiji Izuchi
Journal: Proc. Amer. Math. Soc. 125 (1997), 1153-1159
MSC (1991): Primary 41J35, 46J10
DOI: https://doi.org/10.1090/S0002-9939-97-03884-7
MathSciNet review: 1396982
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\{ T_n \}_n$ be a sequence of bounded linear operators on $L^\infty $ such that $\| T_n \| \to 1$ and $\| T_n g - g \|_\infty \to 0$ for every $g \in QC$. It is proved that $\| T_n f - f \|_\infty \to 0$ for every $f \in L^\infty $.


References [Enhancements On Off] (What's this?)

  • 1. F. Altomare and M. Campiti, Korovkin-type approximation theory and its applications, W. de Gruyter, Berlin and New York, 1994.MR 95g:41001
  • 2. J. Garnett, Bounded analytic functions, Academic Press, New York and London, 1981.MR 83g:30037
  • 3. K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, Englewood Cliffs, N.J., 1962.MR 24:A2844
  • 4. K. Izuchi, $QC$-level sets and quotients of Douglas algebras, J. Funct. Anal. 65 (1986), 293-308.MR 87f:46093
  • 5. K. Izuchi, Countably generated Douglas algebras, Trans. Amer. Math. Soc. 299 (1987), 177-192.MR 88b:46077
  • 6. K. Izuchi, H. Takagi and S. Watanabe, Sequential $BKW$-operators and function algebras, J. Approx. Theory 85(1996), 185-200. MR 97c:46060
  • 7. K. Izuchi, H. Takagi and S. Watanabe, Sequential Korovkin type theorems and weighted composition operators, Acta Sci. Math. (Szeged) 62 (1996), 161-174. CMP 97:02
  • 8. P. P. Korovkin, On convergence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR (N.S) 90 (1953), 961-964. (in Russian)MR 15:236a
  • 9. P. P. Korovkin, Linear operators and approximation theory, Hindustan Publishing Corp., Delhi, 1960.MR 27:561
  • 10. D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405.MR 51:13690
  • 11. D. Sarason, Function theory on the unit circle, Virginia Poly. and State Univ., Blacksburg, 1978.MR 80d:30035
  • 12. E. Scheffold, Über die punktweise konvergenz von operatoren in $C(X)$, Rev. Acad. Ci. Zaragoza 28(1973), 5-12.MR 48:948
  • 13. S. -E. Takahasi, $(T, E)$-Korovkin closures in normed spaces and $BKW$-operators, J. Approx. Theory 82(1995), 340-351.MR 96d:41029
  • 14. D. E. Wulbert, Convergence of operators and Korovkin's theorem, J. Approx. Theory 1 (1968), 381-390.MR 38:3679

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 41J35, 46J10

Retrieve articles in all journals with MSC (1991): 41J35, 46J10


Additional Information

Keiji Izuchi
Affiliation: Department of Mathematics, Niigata University, Niigata 950-21, Japan
Email: izuchi@scux.sc.niigata-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-97-03884-7
Received by editor(s): February 23, 1996
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society