A new type of statistical Cauchy sequence and its relation to Bourbaki completeness

References

• Acar, T., & Mohiuddine, S. A. (2016). Statistical (C,1)(E,1) summability and Korovkin’s theorem. Filomat, 30(2), 361-375. doi:10.2298/FIL1602387A
   Web of Science ®Google Scholar
• Aggarwal, M., & Kundu, S. (2016). More on variants of complete metric spaces. Acta Mathematica Hungarica. doi:10.1007/s10474-016-0682-2
   Web of Science ®Google Scholar
• Aggarwal, M., & Kundu, S. (2017). Boundedness of the relatives of uniformly continuous functions. Topology Proceedings, 49, 105–119.
   Google Scholar
• Aldhaifallah, M., Nisar, K. S., Srivastava, H. M., & Mursaleen, M. (2017). Statistical Λ-convergence in probabilistic normed spaces. Applied Mathematics and Computation, 2017, 7.
   Google Scholar
• Atsuji, M. (1958). Uniform continuity of continuous functions of metric spaces. Pacific Journal of Mathematics, 8, 11–16. doi:10.2140/pjm
   Google Scholar
• Beer, G. (1985). Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance. Proceedings of the American Mathematical Society, 95, 653–658. doi:10.1090/S0002-9939-1985-0810180-3
   Web of Science ®Google Scholar
• Beer, G. (1986). More about metric spaces on which continuous functions are uniformly continuous. Bulletin of the Australian Mathematical Society, 33, 397–406. doi:10.1017/S0004972700003981
   Web of Science ®Google Scholar
• Beer, G. (2008). Between compactness and completeness. Topology and Its Applications, 155, 503–514. doi:10.1016/j.topol.2007.08.020
   Web of Science ®Google Scholar
• Belen, C., & Mohiuddine, S. A. (2013). Generalized weighted statistical convergence and application. Applied Mathematics and Computation, 219(18), 9821–9826.
   Web of Science ®Google Scholar
• Braha, N. L., Srivastava, H. M., & Mohiuddine, S. A. (2014). A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallee Poussin mean. Applied Mathematics and Computation, 228, 162–169.
   Web of Science ®Google Scholar
• Çakall, H. (2009). A study on statistical convergence. Functional Analysis, Approximation and Computation, 1(2), 19–24.
   Google Scholar
• Di Maio, G., & Kočinac, L. D. R. (2008). Statistical convergence in topology. Topology and Its Applications, 156, 28–45. doi:10.1016/j.topol.2008.01.015
   Web of Science ®Google Scholar
• Edely, O. H., Mohiuddine, S. A., & Noman, A. K. (2010). Korovkin type approximation theorems obtained through generalized statistical convergence. Applied Mathematics Letters, 23(11), 1382–1387. doi:10.1016/j.aml.2010.07.004
   Web of Science ®Google Scholar
• Fast, H. (1951). Sur la convergence statistique. Colloquium Mathematicum, 2, 241–244. doi:10.4064/cm-2-3-4-241-244
   Google Scholar
• Fridy, J. A. (1985). On statistical convergence. Analysis, 5, 301–313. doi:10.1524/anly.1985.5.4.301
   Google Scholar
• Fridy, J. A., & Khan, M. K. (1998). Tauberian theorems via statistical convergence. Journal of Mathematical Analysis and Applications, 228, 73–95. doi:10.1006/jmaa.1998.6118
   Web of Science ®Google Scholar
• Fridy, J. A., & Orhan, C. (1993). Lacunary statistical summability. Journal of Mathematical Analysis and Applications, 173, 497–504. doi:10.1006/jmaa.1993.1082
   Web of Science ®Google Scholar
• Gadjiev, A. D., & Orhan, C. (2002). Some approximation theorems via statistical convergence. Rocky Mountain Journal of Mathematics, 32, 129–138. doi:10.1216/rmjm/1030539612
   Web of Science ®Google Scholar
• Garrido, M. I., & Meroño, A. S. (2014). New types of completeness in metric spaces. Annales Academiae Scientiarum Fennicae Mathematica, 39, 733–758. doi:10.5186/aasfm.00
   Web of Science ®Google Scholar
• Jain, T., & Kundu, S. (2007). Atsuji completions: Equivalent characterisations. Topology and Its Applications, 154, 28–38. doi:10.1016/j.topol.2006.03.014
   Web of Science ®Google Scholar
• Kadak, U., Braha, N. L., & Srivastava, H. M. (2017). Statistical weighted B-summability and its applications to approximation theorems. Applied Mathematics and Computation, 302, 80–96.
   Web of Science ®Google Scholar
• Kadak, U., & Mohiuddine, S. A. (2018). Generalized statistically almost convergence based on the difference operator which includes the (p,q)-gamma function and related approximation theorems. Results Mathematical, 73(9). doi:10.1007/s00025-018-0789-6
   Google Scholar
• Kirişci, M., & Karaisa, A. (2017). Fibonacci statistical convergence and Korovkin type approximation theorems. Journal of Inequalities and Applications, 2017(229), 15.
   Google Scholar
• Kováč, E. (2005). On φ-convergence and φ-density. Mathematical Slovaca, 55(3), 329–351.
   Google Scholar
• Küçükaslan, M., Değer, U., & Dovgoshey, O. (2014). On the statistical convergence of metric-valued sequences. Ukrainian Mathematical Journal, 66(5), 796–805. doi:10.1007/s11253-014-0974-z
   Web of Science ®Google Scholar
• Kundu, S., Aggarwal, M., & Hazra, S. (2017). Finitely chainable and totally bounded metric spaces: Equivalent characterizations. Topology and Its Applications, 216, 59–73. doi:10.1016/j.topol.2016.11.008
   Web of Science ®Google Scholar
• Kundu, S., & Jain, T. (2006). Atsuji spaces: Equivalent conditions. Topology Proceedings, 30, 301–325.
   Google Scholar
• Li, K., Lin, S., & Ge, Y. (2015). On statistical convergence in cone metric spaces. Topology and Its Applications, 196, 641–651. doi:10.1016/j.topol.2015.05.038
   Web of Science ®Google Scholar
• Miller, H. I. (1995). A measure theoretical subsequence characterization of statistical convergence. Transactions of the American Mathematical Society, 347, 1811–1819.
   Web of Science ®Google Scholar
• Monteiro, A. A., & Peixoto, M. M. (1951). Le nombre de Lebesgue et la continuité uniforme. Portugaliae Mathematical, 10, 105–113.
   Google Scholar
• Mursaleen, M., & Edely, O. H. H. (2004). Generalized statistical convergence. Information Sciences, 162, 287–294. doi:10.1016/j.ins.2003.09.011
   Web of Science ®Google Scholar
• Nagata, J. (1950). On the uniform topology of bicompactifications. Journal of the Institute of Polytechnics, Osaka City University, 1, 409–422.
   Google Scholar
• Niven, I., Zuckerman, H. S., & Montgomery, H. (1991). An introduction to the theory of numbers (5th ed.). New York, NY: Wiley.
   Google Scholar
• Pehlivan, S., & Mamedov, M. A. (2000). Statistical cluster points and turnpike. Optimization, 48, 93–106. doi:10.1080/02331930008844495
   Google Scholar
• Šalát, T. (1980). On statistically convergent sequences of real numbers. Mathematical Slovaca, 30(2), 139–150.
   Google Scholar
• Schoenberg, I. J. (1959). The integrability of certain functions and related summability methods. The American Mathematical Monthly, 66, 361–775. doi:10.1080/00029890.1959.11989303
   Google Scholar
• Srivastava, H. M., Jena, B. B., Paikray, S. K., & Mishra, U. K. (2018). A certain class of weighted statistical convergence and associated Krovkin-type approximation theorems involving trigonometric functions. Mathematical Methods Applications Sciences, 41, 671–683.
   Web of Science ®Google Scholar
• Steinhaus, H. (1951). Sur la convergence ordinaire et la convergence asymptotique. Colloquium Mathematics, 2, 73–74.
   Google Scholar
• Zygmund, A. (2002). Trigonometric series (3rd ed.). Cambridge, UK: Cambridge University Press.
   Google Scholar