Some aspects of 𝜆-weak convergence using difference operator
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Archana Sharma
und
Vijay Kumar
Abstract
In this paper, we introduce generalized difference weak sequence space classes by utilizing the difference operator
Acknowledgements
We would like to thank the reviewers for their careful reading, which improved the presentation of paper.
References
[1]
B. Altay and F. Başar,
The fine spectrum and the matrix domain of the difference operator Δ on the sequence space
[2] C. Aydın and F. Başar, Some new difference sequence spaces, Appl. Math. Comput. 157 (2004), no. 3, 677–693. 10.1016/j.amc.2003.08.055Suche in Google Scholar
[3] P. Baliarsingh, Some new difference sequence spaces of fractional order and their dual spaces, Appl. Math. Comput. 219 (2013), no. 18, 9737–9742. 10.1016/j.amc.2013.03.073Suche in Google Scholar
[4] F. Başar, Summability Theory and its Applications, 2nd ed., CRC Press, Boca Raton, 2022. 10.1201/9781003294153Suche in Google Scholar
[5] F. Başar and B. Altay, On the space of sequences of 𝑝-bounded variation and related matrix mappings, Ukraïn. Mat. Zh. 55 (2003), no. 1, 108–118; translation in Ukrain. Math. J. 55 (2003), 136–147. Suche in Google Scholar
[6] P. Beuria, P. Baliarsingh and L. Nayak, On a generalized difference sequence and its applications, Proc. Nat. Acad. Sci. India Sect. A 92 (2022), no. 1, 39–46. 10.1007/s40010-020-00689-7Suche in Google Scholar
[7] V. K. Bhardwaj and I. Bala, On weak statistical convergence, Int. J. Math. Math. Sci. 2007 (2007), Article ID 38530. 10.1155/2007/38530Suche in Google Scholar
[8] J. Connor, M. Ganichev and V. Kadets, A characterization of Banach spaces with separable duals via weak statistical convergence, J. Math. Anal. Appl. 244 (2000), no. 1, 251–261. 10.1006/jmaa.2000.6725Suche in Google Scholar
[9] J. S. Connor, The statistical and strong 𝑝-Cesàro convergence of sequences, Analysis 8 (1988), no. 1–2, 47–63. 10.1524/anly.1988.8.12.47Suche in Google Scholar
[10] O. Duman, M. K. Khan and C. Orhan, 𝐴-statistical convergence of approximating operators, Math. Inequal. Appl. 6 (2003), no. 4, 689–699. 10.7153/mia-06-62Suche in Google Scholar
[11] S. Ercan, Y. Altin and Ç. A. Bektaş, On lacunary weak statistical convergence of order 𝛼, Comm. Statist. Theory Methods 49 (2020), no. 7, 1653–1664. 10.1080/03610926.2018.1563185Suche in Google Scholar
[12] A. Esi, B. C. Tripathy and B. Sarma, On some new type generalized difference sequence spaces, Math. Slovaca 57 (2007), no. 5, 475–482. 10.2478/s12175-007-0039-ySuche in Google Scholar
[13] M. Et, On some generalized Cesàro difference sequence spaces, İstanbul Üniv. Fen Fak. Mat. Derg. 55/56 (1996/97), 221–229. Suche in Google Scholar
[14] M. Et and R. Çolak, On some generalized difference sequence spaces, Soochow J. Math. 21 (1995), no. 4, 377–386. Suche in Google Scholar
[15] M. Et, M. Karakaş and V. Karakaya, Some geometric properties of a new difference sequence space defined by de la Vallée–Poussin mean, Appl. Math. Comput. 234 (2014), 237–244. 10.1016/j.amc.2014.01.122Suche in Google Scholar
[16] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244. 10.4064/cm-2-3-4-241-244Suche in Google Scholar
[17] J. A. Fridy, On statistical convergence, Analysis 5 (1985), no. 4, 301–313. 10.1524/anly.1985.5.4.301Suche in Google Scholar
[18] V. Kadets, A. Leonov and C. Orhan, Weak statistical convergence and weak filter convergence for unbounded sequences, J. Math. Anal. Appl. 371 (2010), no. 2, 414–424. 10.1016/j.jmaa.2010.05.031Suche in Google Scholar
[19] H. Kızmaz, On certain sequence spaces, Canad. Math. Bull. 24 (1981), no. 2, 169–176. 10.4153/CMB-1981-027-5Suche in Google Scholar
[20] L. Leindler, Über die verallgemeinerte de la Vallée–Poussinsche Summierbarkeit allgemeiner Orthogonalreihen, Acta Math. Acad. Sci. Hungar. 16 (1965), 375–387. 10.1007/BF01904844Suche in Google Scholar
[21] I. J. Maddox, Elements of Functional Analysis, Cambridge University, Cambridge, 1988. Suche in Google Scholar
[22] I. J. Maddox, Statistical convergence in a locally convex space, Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 1, 141–145. 10.1017/S0305004100065312Suche in Google Scholar
[23] Meenakshi, M. S. Saroa and V. Kumar, Weak statistical convergence defined by de la Vallée–Poussin mean, Bull. Calcutta Math. Soc. 106 (2014), no. 3, 215–224. Suche in Google Scholar
[24] S. A. Mohiuddine, B. Hazarika and H. K. Nashine, Approximation Theory, Sequence Spaces and Applications, Ind. Appl. Math., Springer, Singapore, 2022. 10.1007/978-981-19-6116-8Suche in Google Scholar
[25] M. Mursaleen, 𝜆-statistical convergence, Math. Slovaca 50 (2000), no. 1, 111–115. Suche in Google Scholar
[26] M. Mursaleen and F. Başar, Sequence Spaces: Topics in Modern Summability Theory, Ser. Math. Appl., CRC Press/Taylor, New York, 2020. 10.1201/9781003015116Suche in Google Scholar
[27] F. Nuray, Lacunary weak statistical convergence, Math. Bohem. 136 (2011), no. 3, 259–268. 10.21136/MB.2011.141648Suche in Google Scholar
[28] T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), no. 2, 139–150. Suche in Google Scholar
[29] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361–375. 10.2307/2308747Suche in Google Scholar
[30] A. Sönmez and F. Başar, Generalized difference spaces of non-absolute type of convergent and null sequences, Abstr. Appl. Anal. 2012 (2012), Article ID 435076. 10.1155/2012/435076Suche in Google Scholar
[31] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73–74. 10.4064/cm-2-2-98-108Suche in Google Scholar
[32] B. C. Tripathy, B. Hazarika and B. Choudhary, Lacunary 𝐼-convergent sequences, Kyungpook Math. J. 52 (2012), no. 4, 473–482. 10.5666/KMJ.2012.52.4.473Suche in Google Scholar
[33] T. Yaying, B. Hazarika, S. A. Mohiuddine and M. Et, On sequence spaces due to 𝑙th order 𝑞-difference operator and its spectrum, Iran. J. Sci. 47 (2023), no. 4, 1271–1281. 10.1007/s40995-023-01487-7Suche in Google Scholar
[34] K. Zeller, Theorie der Limitierungsverfahren, Ergeb. Math. Grenzgeb. (3) 15, Springer, Berlin, 1958. 10.1007/978-3-642-52767-8Suche in Google Scholar
[35] A. Zygmund, Trigonometric Series, Cambridge University, London, 2002. 10.1017/CBO9781316036587Suche in Google Scholar
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