DOI QR코드

DOI QR Code

A Note on Spliced Sequences and A-density of Points with respect to a Non-negative Matrix

  • 투고 : 2017.10.12
  • 심사 : 2018.10.10
  • 발행 : 2019.03.23

초록

For $y{\in}{\mathbb{R}}$, a sequence $x=(x_n){\in}{\ell}^{\infty}$, and a non-negative regular matrix A, Bartoszewicz et. al., in 2015, defined the notion of the A-density ${\delta}_A(y)$ of the indices of those $x_n$ that are close to y. Their main result states that if the set of limit points of ($x_n$) is countable and density ${\delta}_A(y)$ exists for any $y{\in}\mathbb{R}$ where A is a non-negative regular matrix, then ${\lim}_{n{\rightarrow}{\infty}}(Ax)_n={\sum}_{y{\in}{\mathbb{R}}}{\delta}_A(y){\cdot}y$. In this note we first show that the result can be extended to a more general class of matrices and then consider a conjecture which naturally arises from our investigations.

키워드

과제정보

연구 과제 주관 기관 : UGC, CSIR

참고문헌

  1. M. Balcerzak, K. Dems and A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl., 328(1)(2007), 715-729. https://doi.org/10.1016/j.jmaa.2006.05.040
  2. A. Bartoszewicz, P. Das and S. Glab, On matrix summability of spliced sequences and A-density of points, Linear Algebra Appl., 487(2015), 22-42. https://doi.org/10.1016/j.laa.2015.08.031
  3. A. Bartoszewicz, S. Glab and A. Wachowicz, Remarks on ideal boundedness, convergence and variation of sequences, J. Math. Anal. Appl., 375(2)(2011), 431-435. https://doi.org/10.1016/j.jmaa.2010.09.023
  4. P. Das, Some further results on ideal convergence in topological spaces, Topology Appl., 159(2012), 2621-2626. https://doi.org/10.1016/j.topol.2012.04.007
  5. H. Fast, Sur la convergence statistique, Colloq. Math., 2(1951), 241-244. https://doi.org/10.4064/cm-2-3-4-241-244
  6. R. Filipow, N. Mrozek, I. Rec law and P. Szuca, Ideal convergence of bounded sequences, J. Symbolic Logic, 72(2)(2007), 501-512. https://doi.org/10.2178/jsl/1185803621
  7. R. Filipow and P. Szuca, On some questions of Drewnowski and Luczak concerning submeasures on ${\mathbb{N}}$, J. Math. Anal. Appl., 371(2)(2010), 655-660. https://doi.org/10.1016/j.jmaa.2010.05.068
  8. R. Filipow and P. Szuca, Density versions of Schur's theorem for ideals generated by submeasures, J. Combin. Theory Ser. A, 117(7)(2010), 943-956. https://doi.org/10.1016/j.jcta.2009.12.005
  9. A. R. Freedman and J. J. Sember, Densities and summability, Pacific J. Math., 95(1981), 293-305. https://doi.org/10.2140/pjm.1981.95.293
  10. J. A. Fridy, On statistical convergence, Analysis, 5(4)(1985), 301-313. https://doi.org/10.1524/anly.1985.5.4.301
  11. R. Henstock, The efficiency of matrices for bounded sequences, J. London Math. Soc., 25(1950), 27-33. https://doi.org/10.1112/jlms/s1-25.1.27
  12. J. Jasinski and I. Reclaw, On spaces with the ideal convergence property, Colloq. Math., 111(1)(2008), 43-50. https://doi.org/10.4064/cm111-1-4
  13. P. Kostyrko, T. Salat and W. Wilczynski, I-convergence, Real Anal. Exchange, 26(2)(2000/2001), 669-685. https://doi.org/10.2307/44154069
  14. B. K. Lahiri and P. Das, I and I*-convergence in topological spaces, Math. Bohem., 130(2005), 153-160. https://doi.org/10.21136/MB.2005.134133
  15. N. Mrozek, Ideal version of Egorov's theorem for analytic P-ideals, J. Math. Anal. Appl., 349(2)(2009), 452-458. https://doi.org/10.1016/j.jmaa.2008.08.032
  16. J. A. Osikiewicz, Summability of spliced sequences, Rocky Mountain J. Math., 35(3)(2005), 977-996. https://doi.org/10.1216/rmjm/1181069717
  17. E. Savas, P. Das and S. Dutta, A note on strong matrix summability via ideals, Appl. Math. Lett., 25(4)(2012), 733-738. https://doi.org/10.1016/j.aml.2011.10.012
  18. I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66(1959), 361-375. https://doi.org/10.1080/00029890.1959.11989303
  19. S. Solecki, Analytic ideals and their applications, Ann. Pure Appl. Logic, 99(1999), 51-72. https://doi.org/10.1016/S0168-0072(98)00051-7