| Title:
|
A Korovkin type approximation theorems via $\scr I$-convergence (English) |
| Author:
|
Duman, O. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
57 |
| Issue:
|
1 |
| Year:
|
2007 |
| Pages:
|
367-375 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Using the concept of $\mathcal {I}$-convergence we provide a Korovkin type approximation theorem by means of positive linear operators defined on an appropriate weighted space given with any interval of the real line. We also study rates of convergence by means of the modulus of continuity and the elements of the Lipschitz class. (English) |
| Keyword:
|
$\scr{I}$-convergence |
| Keyword:
|
positive linear operator |
| Keyword:
|
the classical Korovkin theorem |
| MSC:
|
40A99 |
| MSC:
|
41A10 |
| MSC:
|
41A25 |
| MSC:
|
41A36 |
| idZBL:
|
Zbl 1174.41004 |
| idMR:
|
MR2309970 |
| . |
| Date available:
|
2009-09-24T11:46:13Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128176 |
| . |
| Reference:
|
[1] R. Bojanic, F. Cheng: Estimates for the rate of approximation of functions of bounded variation by Hermite-Fejer polynomials.Proceedings of the Conference of Canadian Math. Soc. 3 (1983), 5–17. MR 0729319 |
| Reference:
|
[2] R. Bojanic, M. K. Khan: Summability of Hermite-Fejer interpolation for functions of bounded variation.J. Nat. Sci. Math. 32 (1992), 5–10. |
| Reference:
|
[3] I. Chlodovsky: Sur la representation des fonctions discontinuous par les polynômes de M. S. Bernstein.Fund. Math. 13 (1929), 62–72. 10.4064/fm-13-1-62-72 |
| Reference:
|
[4] J. S. Connor: On strong matrix summability with respect to a modulus and statistical convergence.Canad. Math. Bull. 32 (1989), 194–198. Zbl 0693.40007, MR 1006746, 10.4153/CMB-1989-029-3 |
| Reference:
|
[5] O. Duman, C. Orhan: Statistical approximation by positive linear operators.Stud. Math. 161 (2004), 187–197. MR 2033235, 10.4064/sm161-2-6 |
| Reference:
|
[6] O. Duman, M. K. Khan, C. Orhan: $A$-statistical convergence of approximating operators.Math. Inequal. Appl. 4 (2003), 689–699. MR 2013529 |
| Reference:
|
[7] H. Fast: Sur la convergence statistique.Colloq. Math. 2 (1951), 241–244. Zbl 0044.33605, MR 0048548, 10.4064/cm-2-3-4-241-244 |
| Reference:
|
[8] A. R. Freedman, J. J. Sember: Densities and summability.Pacific J. Math. 95 (1981), 293–305. MR 0632187, 10.2140/pjm.1981.95.293 |
| Reference:
|
[9] J. A. Fridy: On statistical convergence.Analysis 5 (1985), 301–313. Zbl 0588.40001, MR 0816582 |
| Reference:
|
[10] J. A. Fridy, H. I. Miller: A matrix characterization of statistical convergence.Analysis 11 (1991), 59–66. MR 1113068, 10.1524/anly.1991.11.1.59 |
| Reference:
|
[11] G. H. Hardy: Divergent Series.Oxford Univ. Press, London, 1949. Zbl 0032.05801, MR 0030620 |
| Reference:
|
[12] M. K. Khan, B. Della Vecchia, A. Fassih: On the monotonicity of positive linear operators.J. Approximation Theory 92 (1998), 22–37. MR 1492856, 10.1006/jath.1996.3113 |
| Reference:
|
[13] E. Kolk: Matrix summability of statistically convergent sequences.Analysis 13 (1993), 77–83. Zbl 0801.40005, MR 1245744, 10.1524/anly.1993.13.12.77 |
| Reference:
|
[14] P. P. Korovkin: Linear Operators and Theory of Approximation.Hindustan Publ. Comp., Delhi, 1960. MR 0150565 |
| Reference:
|
[15] P. Kostyrko, T. Šalát, W. Wilczyński: ${I}$-convergence.Real Anal. Exchange 26 (2000/01), 669–685. MR 1844385 |
| Reference:
|
[16] C. Kuratowski: Topologie I.PWN, Warszawa, 1958. |
| Reference:
|
[17] H. I. Miller: A measure theoretical subsequence characterization of statistical convergence.Trans. Amer. Math. Soc. 347 (1995), 1811–1819. Zbl 0830.40002, MR 1260176, 10.1090/S0002-9947-1995-1260176-6 |
| Reference:
|
[18] I. Niven, H. S. Zuckerman, H. Montgomery: An Introduction to the Theory of Numbers. 5th Edition.Wiley, New York, 1991. MR 1083765 |
| Reference:
|
[19] H. L. Royden: Real Analysis. 2nd edition.Macmillan Publ., New York, 1968. MR 1013117 |
| . |
