Korovkin-type approximation theorem for Bernstein Stancu operator of rough statistical convergence of triple sequence
DOI:
https://doi.org/10.5269/bspm.v38i7.45874Abstract
We obtain a Korovkin-type approximation theorem for Bernstein Stancu polynomials of rough statistical convergence of triple sequences of positive linear operators of three variables from $H_{\omega}\left( K\right) $ to $C_{B}\left( K\right) $, where $K=[0,\infty)\times\lbrack0,\infty )\times\lbrack0,\infty)$ and $\omega$ is non-negative increasing function on $K$.
References
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27. S. A. Mohiuddine and A. Alotaibi, Statistical convergence and approximation theorems for functions of two variables, J. Comput. Anal. Appl., 15(2), 218-223 (2013).
28. S. A. Mohiuddine and A. Alotaibi, Korovkin second theorem via statistical summability (C, 1), J. Inequa. Appl., 2013, Article 149, 9 pages (2013). https://doi.org/10.1186/1029-242X-2013-149
29. M. Mursaleen and A. Kilicman, Korovkin Second Theorem via B-Statistical A-Summability, Abstr. Appl. Anal., (2013). https://doi.org/10.1155/2013/598963
30. M. Mursaleen, F. Khan, A. Khan and A. Kilicman, Some approximation results for generalized Kantorovich-type operators. J. Inequal. Appl., 2013:585, 1-14 (2013). https://doi.org/10.1186/1029-242X-2013-585
31. H. X. Phu, Rough convergence in normed linear spaces, Numer. Funct. Anal. Optimiz., 22, 199-222 (2001).
https://doi.org/10.1081/NFA-100103794
32. H. X. Phu, Rough continuity of linear operators, Numer. Funct. Anal. Optimiz., 23, 139-146 (2002).
https://doi.org/10.1081/NFA-120003675
33. H. X. Phu, Rough convergence in infinite dimensional normed spaces, Numer. Funct. Anal. Optimiz., 24, 285-301 (2003). https://doi.org/10.1081/NFA-120022923
34. A. Sahiner, M. Gurdal and F. K. Duden, Triple sequences and their statistical convergence, Selcuk J. Appl. Math., 8(2), 49-55 (2007).
35. A. Sahiner and B. C. Tripathy, Some I related properties of triple sequences, Selcuk J. Appl. Math., 9(2), 9-18 (2008).
36. T. Salat, On statistical convergence of real numbers, Math. Slovaca, 30, 139-150 (1980).
37. E. Savas and A. Esi, Statistical convergence of triple sequences on probabilistic normed space, Ann. Uni. Craiova, Math. Compu. Sci. Series, 39(2), 226-236 (2012).
38. I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66, 361-375 (1959). https://doi.org/10.1080/00029890.1959.11989303
39. N. Subramanian and A. Esi, On triple sequence space of Bernstein operator of 3 of rough -statistical convergence in probability defined by Musielak-Orlicz function of p-metric, Electron. J. Math. Anal. Appl., 6(1), 198-203 (2018).
40. B. C. Tripathy and R. Goswami, On triple difference sequences of real numbers in probabilistic normed spaces, Proyecciones J. Math., 33(2), 157-174 (2014). https://doi.org/10.4067/S0716-09172014000200003
41. Ayten Esi, M. Kemal Ozdemir and N. Subramanian, Korovkin-type approximation theorem for Bernstein Stancu operator of rough statistical convergence of triple sequence, 2nd International Conference of Mathematical Sciences, 31 July 2018-6 August 2018 (ICMS 2018) Maltepe University, Istanbul, Turkey.
2. F. Altomare and M. Campiti, Korovkin type approximation theory and its applications, Walter de Gruyter Publ., Berlin (1994). https://doi.org/10.1515/9783110884586
3. S. Aytar, Rough statistical convergence, Numer. Funct. Anal. Optimiz., 29(3-4), 291-303 (2008). https://doi.org/10.1080/01630560802001064
4. S. Aytar, The rough limit set and the core of a real sequence, Numer. Funct. Anal. Optimiz., 29(3-4), 283-290 (2008). https://doi.org/10.1080/01630560802001056
5. N. L. Braha, H. M. Srivastava and S. A. Mohiuddine, A Korovkin's type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallee Poussin mean, Appl. Math. Comput., 228, 162-169 (2014). https://doi.org/10.1016/j.amc.2013.11.095
6. S. Debnath, B. Sarma and B.C. Das, Some generalized triple sequence spaces of real numbers, J. Nonlinear Anal. Optimiz., 6(1), 71-79 (2015).
7. O. Duman, A Korovkin type approximation throrems via I-convergence, Czechoslovak Math. J., 75(132), 367-375 (2007). https://doi.org/10.1007/s10587-007-0065-5
8. O. Duman, M. K. Khan and C. Orhan, A-statistical convergence of approximating operators, Math. Inequa. Appl., 6(4), 689-699 (2003). https://doi.org/10.7153/mia-06-62
9. E. Dundar and C. Cakan, Rough convergence of double sequences, Gulf J. Math., 2(1), 45-51 (2014).
10. A. J. Dutta, A. Esi and B.C. Tripathy, Statistically convergent triple sequence spaces defined by Orlicz function, J. Math. Anal., 4(2), 16-22 (2013).
11. O. H. H. Edely, S. A. Mohiuddine and A. Noman, Korovkin type approximation theorems obtained through generalized statistical convergence, Appl. Math. Letters, 23(11), 1382-1387 (2010). https://doi.org/10.1016/j.aml.2010.07.004
12. A. Esi, On some triple almost lacunary sequence spaces defined by Orlicz functions, Research and Reviews: Discrete Math. Structures, 1(2), 16-25 (2014).
13. A. Esi and M. Necdet Catalbas, Almost convergence of triple sequences, Global J. Math. Anal., 2(1), 6-10 (2014).
https://doi.org/10.14419/gjma.v2i1.1709
14. A. Esi and E. Savas, On lacunary statistically convergent triple sequences in probabilistic normed space, Appl. Math. Inf. Sci., 9(5), 2529-2534 (2015).
15. A. Esi and N. Subramanian, On triple sequence spaces of Bernstein operator of 3 of rough - statistical convergence in probability of random variables defined by Musielak-Orlicz function, Int. J. Open Probl. Comput. Sci. Math., 11(2), 62-70 (2018). https://doi.org/10.12816/0049063
16. A. Esi and N. Subramanian, Some triple difference rough Cesaro and lacunary statistical sequence spaces, J. Math. Appl., 41, 1-13 (2018). https://doi.org/10.7862/rf.2018.7
17. A. Esi, S. Araci and M. Acikgoz, Statistical Convergence of Bernstein Operators, Appl. Math. Inf. Sci., 10(6), 2083-2086 (2016). https://doi.org/10.18576/amis/100610
18. A. Esi, 3-Statistical convergence of triple sequences on probabilistic normed space, Global J. Math. Anal., 1(2), 29-36 (2013). https://doi.org/10.14419/gjma.v1i2.885
19. H. Fast, Sur la convergence statistique, Colloq. Math., 2, 241-244 (1951). https://doi.org/10.4064/cm-2-3-4-241-244
20. J. A. Fridy, On statistical convergence, Analysis, 5, 301-313 (1985). https://doi.org/10.1524/anly.1985.5.4.301
21. A. D. Gadjiev, The convergence problem for a sequence of positive linear operators an unbounded sets, and theorems analogous to that of P.P. Korovkin, Soviet Math. Dokl., 15, 1433-1436 (1974).
22. A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32(1), 129-138 (2002). https://doi.org/10.1216/rmjm/1030539612
23. B. Hazarika, N. Subramanian and A. Esi, On rough weighted ideal convergence of triple sequence of Bernstein polynomials, Proccedings of the Jangjeon Mathematical Society, 21(3), 497-506 (2018).
24. P. P. Korovkin, Linear operators and the theory of approximation, India, Delhi (1960).
25. P. Malik and M. Maity, On rough statistical convergence of double sequences in normed linear spaces, Afr. Mat., 27(1), 141-148 (2016). https://doi.org/10.1007/s13370-015-0332-9
26. S. A. Mohiuddine, An application of almost convergence in approximation theorems, Appl. Math. Letters, 24(11), 1856-1860 (2011). https://doi.org/10.1016/j.aml.2011.05.006
27. S. A. Mohiuddine and A. Alotaibi, Statistical convergence and approximation theorems for functions of two variables, J. Comput. Anal. Appl., 15(2), 218-223 (2013).
28. S. A. Mohiuddine and A. Alotaibi, Korovkin second theorem via statistical summability (C, 1), J. Inequa. Appl., 2013, Article 149, 9 pages (2013). https://doi.org/10.1186/1029-242X-2013-149
29. M. Mursaleen and A. Kilicman, Korovkin Second Theorem via B-Statistical A-Summability, Abstr. Appl. Anal., (2013). https://doi.org/10.1155/2013/598963
30. M. Mursaleen, F. Khan, A. Khan and A. Kilicman, Some approximation results for generalized Kantorovich-type operators. J. Inequal. Appl., 2013:585, 1-14 (2013). https://doi.org/10.1186/1029-242X-2013-585
31. H. X. Phu, Rough convergence in normed linear spaces, Numer. Funct. Anal. Optimiz., 22, 199-222 (2001).
https://doi.org/10.1081/NFA-100103794
32. H. X. Phu, Rough continuity of linear operators, Numer. Funct. Anal. Optimiz., 23, 139-146 (2002).
https://doi.org/10.1081/NFA-120003675
33. H. X. Phu, Rough convergence in infinite dimensional normed spaces, Numer. Funct. Anal. Optimiz., 24, 285-301 (2003). https://doi.org/10.1081/NFA-120022923
34. A. Sahiner, M. Gurdal and F. K. Duden, Triple sequences and their statistical convergence, Selcuk J. Appl. Math., 8(2), 49-55 (2007).
35. A. Sahiner and B. C. Tripathy, Some I related properties of triple sequences, Selcuk J. Appl. Math., 9(2), 9-18 (2008).
36. T. Salat, On statistical convergence of real numbers, Math. Slovaca, 30, 139-150 (1980).
37. E. Savas and A. Esi, Statistical convergence of triple sequences on probabilistic normed space, Ann. Uni. Craiova, Math. Compu. Sci. Series, 39(2), 226-236 (2012).
38. I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66, 361-375 (1959). https://doi.org/10.1080/00029890.1959.11989303
39. N. Subramanian and A. Esi, On triple sequence space of Bernstein operator of 3 of rough -statistical convergence in probability defined by Musielak-Orlicz function of p-metric, Electron. J. Math. Anal. Appl., 6(1), 198-203 (2018).
40. B. C. Tripathy and R. Goswami, On triple difference sequences of real numbers in probabilistic normed spaces, Proyecciones J. Math., 33(2), 157-174 (2014). https://doi.org/10.4067/S0716-09172014000200003
41. Ayten Esi, M. Kemal Ozdemir and N. Subramanian, Korovkin-type approximation theorem for Bernstein Stancu operator of rough statistical convergence of triple sequence, 2nd International Conference of Mathematical Sciences, 31 July 2018-6 August 2018 (ICMS 2018) Maltepe University, Istanbul, Turkey.
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2019-10-14
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