In this work, we investigate a new type of convergence known as relative statistical convergence through the use of the deferred Nörlund and deferred Riesz means. We demonstrate that the idea of deferred Nörlund and deferred Riesz statistically relative uniform convergence is significantly stronger than deferred Nörlund and deferred Riesz statistically uniform convergence. We provide some interesting examples which explain the validity of the theoretical results and effectiveness of constructed sequence spaces. Furthermore, as an application point of view we prove the Korovkin-type approximation theorem in the context of relative equi-statistical convergence for real valued functions and demonstrate that our theorem effectively extends and most of the earlier existing results. Finally, we present an example involving the Meyer-König-Zeller operator of real sequences proving that our theorem is a stronger approach than its classical and statistical version.
Citation: Lian-Ta Su, Kuldip Raj, Sonali Sharma, Qing-Bo Cai. Applications of relative statistical convergence and associated approximation theorem[J]. AIMS Mathematics, 2022, 7(12): 20838-20849. doi: 10.3934/math.20221142
In this work, we investigate a new type of convergence known as relative statistical convergence through the use of the deferred Nörlund and deferred Riesz means. We demonstrate that the idea of deferred Nörlund and deferred Riesz statistically relative uniform convergence is significantly stronger than deferred Nörlund and deferred Riesz statistically uniform convergence. We provide some interesting examples which explain the validity of the theoretical results and effectiveness of constructed sequence spaces. Furthermore, as an application point of view we prove the Korovkin-type approximation theorem in the context of relative equi-statistical convergence for real valued functions and demonstrate that our theorem effectively extends and most of the earlier existing results. Finally, we present an example involving the Meyer-König-Zeller operator of real sequences proving that our theorem is a stronger approach than its classical and statistical version.
[1] | F. Altomare, Korovkin-type theorems and approximation by positive linear operator, Surveys in Approximation Theory, 5 (2010), 92–164. |
[2] | M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl., 328 (2007), 715–729. https://doi.org/10.1016/j.jmaa.2006.05.040 doi: 10.1016/j.jmaa.2006.05.040 |
[3] | J. Connor, P. Leonetti, A characterization of $(\mathcal{I}, \mathcal{J})-$regular matrices, J. Math. Anal. Appl., 504 (2021), 125374. https://doi.org/10.1016/j.jmaa.2021.125374 doi: 10.1016/j.jmaa.2021.125374 |
[4] | E. W. Chittenden, On the limit functions of sequences of continuous functions converging relatively uniformly, Trans. Amer. Math. Soc., 20 (1919), 179–184. https://doi.org/10.1090/S0002-9947-1919-1501120-6 doi: 10.1090/S0002-9947-1919-1501120-6 |
[5] | E. W. Chittenden, Relatively uniform convergence of sequences of functions, Trans. Amer. Math. Soc., 15 (1914), 197–201. https://doi.org/10.1090/S0002-9947-1914-1500972-9 doi: 10.1090/S0002-9947-1914-1500972-9 |
[6] | J. S. Connor, The statistical and strong $p$-Cesàro convergence of sequence, Analysis, 8 (1988), 47–63. https://doi.org/10.1524/anly.1988.8.12.47 doi: 10.1524/anly.1988.8.12.47 |
[7] | H. Dutta, S. K. Paikray, B. B. Jena, On statistical deferred Cesàro summability, In: Current trends in mathematical analysis and its interdisciplinary applications, Cham: Birkhäuser, 2019,885–909. https://doi.org/10.1007/978-3-030-15242-0_23 |
[8] | K. Demirci, S. Orhan, Statistically relatively uniform convergence of positive linear operators, Results Math., 69 (2016), 359–367. https://doi.org/10.1007/s00025-015-0484-9 doi: 10.1007/s00025-015-0484-9 |
[9] | K. Demirci, F. Dirik, S. Yildiz, Deferred Nörlund statistical relative uniform convergence and Korovkin-type approximation Theorem, Commun. Fsc. Sci. Univ. Ank. Ser. A1 Math. Stat., 70 (2021), 279–289. https://doi.org/10.31801/CFSUASMAS.807169 doi: 10.31801/CFSUASMAS.807169 |
[10] | M. Et, B. C. Tripathy, A. J. Dutta, On pointwise statistical convergence of order $\alpha$ of sequences of fuzzy mappings, Kuwait J. Sci., 41 (2014), 17–30. |
[11] | J. A. Fridy, On statistical convergent, Analysis, 5 (1985), 301–313. https://doi.org/10.1524/anly.1985.5.4.301 doi: 10.1524/anly.1985.5.4.301 |
[12] | J. A. Fridy, Statistical limit points, Proc. Amer. Math. Soc., 118 (1993), 1187–1192. https://doi.org/10.1090/S0002-9939-1993-1181163-6 doi: 10.1090/S0002-9939-1993-1181163-6 |
[13] | H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244. https://doi.org/10.4064/CM-2-3-4-241-244 doi: 10.4064/CM-2-3-4-241-244 |
[14] | A. Gökhan, M. Güngör, M. Et, Statistical convergence of double sequences of real-valued functions, Int. Math. Forum, 2 (2007), 365-374. http://doi.org/10.12988/imf.2007.07033 doi: 10.12988/imf.2007.07033 |
[15] | B. B. Jena, S. K. Paikray, S. A. Mohiuddine, V. N. Mishra, Relatively equi-statistical convergence via deferred Nörlund mean based on difference operator of fractional-order and related approximation theorems, AIMS Mathematics, 5 (2020), 650–672. https://doi.org/10.3934/math.2020044 doi: 10.3934/math.2020044 |
[16] | S. Jasrotia, U. P. Singh, K. Raj, Application of statistical convergence of order $(\eta, \delta +\gamma)$ in difference sequence spaces of fuzzy numbers, J. Intell. Fuzzy Syst., 40 (2021), 4695–4703. https://doi.org/10.3233/JIFS-201539 doi: 10.3233/JIFS-201539 |
[17] | S. Karakuş, K. Demirci, O. Duman, Equi-statistical Convergence of positive linear operators, J. Math. Anal. Appl., 339 (2008), 1065–1072. https://doi.org/10.1016/j.jmaa.2007.07.050 doi: 10.1016/j.jmaa.2007.07.050 |
[18] | S. A. Mohiuddine, A. Alotaibi, M. Mursaleen, Statistical summability $(C, 1)$ and a Korovkin type approximation theorem, J. Inequal. Appl., 2012 (2012), 172. https://doi.org/10.1186/1029-242X-2012-172 doi: 10.1186/1029-242X-2012-172 |
[19] | E. H. Moore, An introduction to a form of general analysis, New Haven: Yale University Press, 1910. |
[20] | G. M. Petersen, Regular matrix transformations, London: McGraw-Hill, 1996. |
[21] | K. Raj, A. Choudhary, Relative modular uniform approximation by means of power series method with applications, Revista de la Unión Matemática Argentina, 60 (2019), 187–208. https://doi.org/10.33044/revuma.v60n1a11 doi: 10.33044/revuma.v60n1a11 |
[22] | K. Saini, K. Raj, Application of statistical convergence in complex uncertain sequence via deferred Reisz mean, Int. J. Uncertain Fuzz. Knowl. Based Syst., 29 (2021), 337–351. https://doi.org/10.1142/S021848852150015X doi: 10.1142/S021848852150015X |
[23] | E. Savas, M. Mursaleen, On statistically convergent double sequence of fuzzy numbers, Inform. Sci., 162 (2004), 183–192. https://doi.org/10.1016/j.ins.2003.09.005 doi: 10.1016/j.ins.2003.09.005 |
[24] | P. D. Srivastava, S. Ojha, $\lambda-$Statistical convergence of fuzzy numbers and fuzzy functions of order $\theta$, Soft Comput., 18 (2014), 1027–1032. https://doi.org/10.1007/s00500-013-1125-4 doi: 10.1007/s00500-013-1125-4 |
[25] | T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30 (1980), 139–150. |
[26] | I. J. Schoenberg, The integrability of certain function and related summability methods, Amer. Math. Mon., 66 (1959), 361–375. https://doi.org/10.1080/00029890.1959.11989303 doi: 10.1080/00029890.1959.11989303 |
[27] | B. C. Tripathy, A. Baruah, M. Et, M. Gungor, On almost statistical convergence of new type of generalized difference sequence of fuzzy numbers, Iran. J. Sci. Technol. A, 36 (2012), 147–155. |