Characterization of rough weighted statistical limit set
-
Pratulananda Das
,
Sanjoy Ghosal
Abstract
Our focus is to generalize the definition of the weighted statistical convergence in a wider range of the weighted sequence {tn}n∈ℕ. We extend the concept of weighted statistical convergence and rough statistical convergence to renovate a new concept namely, rough weighted statistical convergence. On a continuation we also define rough weighted statistical limit set. In the year (2008) Aytar established the following results:
The diameter of rough statistical limit set of a real sequence is ≤ 2r (where r is the degree of roughness) and in general it has no smaller bound.
If the rough statistical limit set is non-empty then the sequence is statistically bounded.
If x∗ and c belong to rough statistical limit set and statistical cluster point set respectively, then |x∗ − c| ≤ r.
We investigate whether the above mentioned three results are satisfied for rough weighted statistical limit set or not?
Answer is no.
So our main objective is to interpret above mentioned different behaviors of the new convergence and characterize the rough weighted statistical limit set. Also we show that this set satisfies some topological properties like boundedness, compactness, path connectedness etc.
Communicated by Ján Borsík
Acknowledgement
Thankful to the Editor and Referees for their several valuable suggestions.
References
[1] Aytar, S.: The rough limit set and the core of a real sequence, Numer. Funct. Anal. Optim. 29(3–4) (2008), 283–290.10.1080/01630560802001056Search in Google Scholar
[2] Aytar, S.: Rough statistical convergence, Numer. Funct. Anal. Optim. 29(3–4) (2008), 291–303.10.1080/01630560802001064Search in Google Scholar
[3] Gecit Akçay, F.—Aytar, S.: Rough convergence of a sequence of fuzzy numbers, Bull. Math. Anal. Appl. 7(4) (2015), 17–23.Search in Google Scholar
[4] Činčura, J.—Šalát, T.—Sleziak, M.—Toma, V.: Sets of statistical cluster points and 𝓘-cluster points, Real Anal. Exchange 30 (2004/2005), 565–580.10.14321/realanalexch.30.2.0565Search in Google Scholar
[5] Connor, S. J.: The statistical and strong p-Cesaro convergence of sequences, Analysis 8(1–2) (1988), 47–63.10.1524/anly.1988.8.12.47Search in Google Scholar
[6] Das, P.—Kostyrko, P.—Wilczynski, W—Malik, P.: I and I∗-convergence of double sequences, Math. Slovaca 58 (2008), 605–620.10.2478/s12175-008-0096-xSearch in Google Scholar
[7] Das, P.—Ghosal, S.—Pal, S: Extending asymmetric convergence and Cauchy condition using ideals, Math. Slovaca 63 (2013), 545–562.10.2478/s12175-013-0117-2Search in Google Scholar
[8] Das, P.—Ghosal, S.—Ghosh, A.: Rough statistical convergence of a sequence of random variables in probability, Afr. Mat. 26 (2015), 1399–1412.10.1007/s13370-014-0295-2Search in Google Scholar
[9] Fast, H.: Sur la convergence statistique, Colloquium Math. 2 (1951), 241–244.10.4064/cm-2-3-4-241-244Search in Google Scholar
[10] Freedman, R. A.—Sember, J. J.—Raphael, M.: Some Cesàro-type summability space, Proc. London Math. Soc. 37(3) (1978), 508–520.10.1112/plms/s3-37.3.508Search in Google Scholar
[11] Fridy, J. A.: On statistical convergence, Analysis 5(4) (1985), 301–313.10.1524/anly.1985.5.4.301Search in Google Scholar
[12] Ghosal, S.: Statistical convergence of a sequence of random variables and limit theorems, Appl. Maths. 58(4) (2013), 423–437.10.1007/s10492-013-0021-7Search in Google Scholar
[13] Ghosal, S.: Generalized weighted random convergence in probability, Appl. Math. Comput. 249 (2014), 502–509.10.1016/j.amc.2014.10.056Search in Google Scholar
[14] Ghosal, S.: Weighted statistical convergence of order α and its applications, J. Egyptian Math. Soc. 24(1) (2016), 60–67.10.1016/j.joems.2014.08.006Search in Google Scholar
[15] Ghosal, S.—Banerjee, M.—Ghosh, A.: Weighted modulus Sθ-convergence of order α in probability, Arab. J. Math. Sci. 23 (2017), 242–257.10.1016/j.ajmsc.2017.03.001Search in Google Scholar
[16] Kostyrko, P.—Mačaj, M.—Šalát, T.—Strauch, O.: On statistical limit points, Proc. Amer. Math. Soc. 129(9) (2001), 2647–2654.10.1090/S0002-9939-00-05891-3Search in Google Scholar
[17] Kostyrko, P.—Mačaj, M.—Sleziak, M.: I-convergence and extremal I-limit points, Math. Slovaca 55 (2005), 443–464.Search in Google Scholar
[18] Karakaya, V.—Chishti, T. A.: Weighted statistical convergence, Iranian J. Sci. Technol.Trans. A 33(A3) (2009), 219–223.Search in Google Scholar
[19] Mursaleen, M.—Alotaibi, A.: On I-convergence in random 2-normed spaces, Math. Slovaca 61 (2011), 933–940.10.2478/s12175-011-0059-5Search in Google Scholar
[20] Mursaleen, M.—Karakaya, V.—Erturk, M.—Gursoy, F.: Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl. Math. Comput. 218(18) (2012), 9132–9137.10.1016/j.amc.2012.02.068Search in Google Scholar
[21] Mursaleen, M.—Mohiuddine, A. S.: On ideal convergence in probabilistic normed spaces, Math. Slovaca 62 (2012), 49–62.10.2478/s12175-011-0071-9Search in Google Scholar
[22] Pehlivan, S.—Mamedov, A. M.: Statistical cluster points and turnpike, Optimization 48(1) (2000), 93–106.10.1080/02331930008844495Search in Google Scholar
[23] Phu, X. H.: Rough convergence in normed linear spaces, Numer. Funct. Anal. Optim. 22(1–2) (2001), 199–222.10.1081/NFA-100103794Search in Google Scholar
[24] Phu, X. H.: Rough continuity of linear operators, Numer. Funct. Anal. Optim. 23(1–2) (2002), 139–146.10.1081/NFA-120003675Search in Google Scholar
[25] Phu, X. H.: Rough convergence in infinite dimensional normed space, Numer. Funct. Anal. Optimiz. 24(3–4) (2003), 285–301.10.1081/NFA-120022923Search in Google Scholar
[26] Phu, X. H.—Truong, T. V.: Invariant property of rough contractive mappings, Vietnam J. Math. 28 (2000), 275–290.Search in Google Scholar
[27] Rath, D.—Tripathy, B. C.: On statistically convergent and statistically Cauchy sequences, Indian J. Pure and Appl. Math. 25(4) (1994), 381–386.Search in Google Scholar
[28] Šalát, T.: On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139–150.Search in Google Scholar
[29] Schonenberg, J. I.: The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361–375.10.1080/00029890.1959.11989303Search in Google Scholar
[30] Steinhaus, H.: Sur la convergence ordinaire et la convergence asymptotique, Colloquium Math. 2 (1951), 73–74.Search in Google Scholar
[31] Tripathy, B. C.: On statistically convergent and statistically bounded sequences, Bull. Malays. Math. Soc. (Second Series) 20 (1997), 31–33.Search in Google Scholar
[32] Tripathy, B. C.—Hazarika, B: Paranorm I-convergent sequence spaces, Math. Slovaca 59 (2009), 485–494.10.2478/s12175-009-0141-4Search in Google Scholar
© 2018 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- A multi-parameter generalization of the symmetric algorithm
- Single identities forcing lattices to be Boolean
- Weighted uniform density ideals
- A generalized class of restricted Stirling and Lah numbers
- The Riemann hypothesis and universality of the Riemann zeta-function
- Distance functions on the sets of ordinary elliptic curves in short Weierstrass form over finite fields of characteristic three
- The drazin inverse of the sum of two matrices
- Refinements of the majorization-type inequalities via green and fink identities and related results
- Characterizations and properties of graphs of Baire functions
- Some improvements of the young mean inequality and its reverse
- On characteristic of bounded analytic functions involving hyperbolic derivative
- A uniqueness problem for entire functions related to Brück’s conjecture
- System of nonlocal resonant boundary value problems involving p-Laplacian
- Construction of a unique mild solution of one-dimensional Keller-Segel systems with uniformly elliptic operators having variable coefficients
- Infinitely many solutions for non-homogeneous Neumann problems in Orlicz-Sobolev spaces
- Characterization of rough weighted statistical limit set
- Approximation by Baskakov-Durrmeyer operators based on (p, q)-integers
- On generalized 4-th root metrics of isotropic scalar curvature
- Compactifications from generators and relations
- Diophantine quadruples with values in k-generalized Fibonacci numbers
Articles in the same Issue
- A multi-parameter generalization of the symmetric algorithm
- Single identities forcing lattices to be Boolean
- Weighted uniform density ideals
- A generalized class of restricted Stirling and Lah numbers
- The Riemann hypothesis and universality of the Riemann zeta-function
- Distance functions on the sets of ordinary elliptic curves in short Weierstrass form over finite fields of characteristic three
- The drazin inverse of the sum of two matrices
- Refinements of the majorization-type inequalities via green and fink identities and related results
- Characterizations and properties of graphs of Baire functions
- Some improvements of the young mean inequality and its reverse
- On characteristic of bounded analytic functions involving hyperbolic derivative
- A uniqueness problem for entire functions related to Brück’s conjecture
- System of nonlocal resonant boundary value problems involving p-Laplacian
- Construction of a unique mild solution of one-dimensional Keller-Segel systems with uniformly elliptic operators having variable coefficients
- Infinitely many solutions for non-homogeneous Neumann problems in Orlicz-Sobolev spaces
- Characterization of rough weighted statistical limit set
- Approximation by Baskakov-Durrmeyer operators based on (p, q)-integers
- On generalized 4-th root metrics of isotropic scalar curvature
- Compactifications from generators and relations
- Diophantine quadruples with values in k-generalized Fibonacci numbers
