On statistical type convergence in uniform spaces

Document Type : Research Paper


1 Department of‎ ‎Non-harmonic analysis‎, ‎Institute of Mathematics and Mechanics of NAS of Azerbaijan‎, ‎9‎, ‎B.Vahabzade Str.‎, ‎AZ 1141‎, ‎Baku‎, ‎Azerbaijan.

2 Department of‎ ‎Non-harmonic analysis‎, ‎Institute of Mathematics and Mechanics of NAS of Azerbaijan‎, ‎9‎, ‎B‎. ‎Vahabzade Str.‎, ‎AZ 1141‎, ‎Baku‎, ‎Azerbaijan.


The concept of ${\mathscr{F}}_{st}$-fundamentality is introduced in uniform spaces, generated by some filter ${\mathscr{F}}$. Its equivalence to the concept of ${\mathscr{F}}$-convergence in uniform spaces is proved. This convergence generalizes many kinds of convergence, including the well-known statistical convergence.


Main Subjects

S. Aytar and S. Pehlivan, Statistically monotonic and statistically bounded sequences of fuzzy numbers, Inform. Sci. 176, (2006), no. 6, 734--744.
N. Bourbaki, General Topology, Nauka, Moscow, 1968.
T. C. Brown and A. R. Freedman, The uniform density of sets of integers and Fermat's last Theorem, C. R. Math. Rep. Acad. Sci. Canada 12 (1990), no. 1, 1--6.
H. Cakalli, Lacunary statistical convergence in topological groups, Indian J. Pure Appl. Math. 26 (1995), no. 2, 113--119.
J. S. Connor, Two valued measures and summability, Analysis 10 (1990), no. 4, 373--385.
J. Connor, The statistical and strong p-Cesro convergence of sequences, Analysis 8 (1988), no. 1-2, 47--63.
A. J. Dutta and B. C. Tripathy, Statistically pre-Cauchy fuzzy real-valued sequences defined by Orlicz function, Proyecciones 33 (2014), no. 3, 235--243.
R. Edwards, Functional Analysis, Theory and Applications, Holt, Rinehart and Winston, New York-Toronto-London 1965.
R. Erdos and G. Tenenbaum, Sur les densits de certaines suites d'entiers, Proc. London Math. Soc. 59 (1989), no. 3, 417--438.
H. Fast, Sur la convergence statistique, (French) Colloquium Math. 2 (1951), 241--244.
J. A. Fridy and M. K. Khan, Tauberian theorems via statistical convergence, J. Math. Anal. Appl. 228 (1998), no. 1, 73--95.
A. R. Freedman and J. J. Sember, Densities and summability, Pacific J. Math. 95 (1981), no. 2, 293--305.
J. A. Fridy, On statistical convergence, Analysis 5 (1985), no. 4, 301--313.
A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32, (2002), no. 1, 129--138.
A. D. Gadjiev Simultaneous statistical approximation of analytic functions and their derivatives by k-positive linear operators, Azerb. J. of Math. 1 (2011), no. 1, 57--66.
J. L. Kelley, General topology, TLNY, 1957.
P. Kostyrko, W. Wilczynski and T. Salat, I-convergence, Real Anal. Exchange 26 (2000), no. 2, 669--686.
P. Kostyrko, M. Macaj and T. Salat, Statistical convergence and I-convergence, Real Analysis Exchange, 1999.
M. Kuchukaslan, Deger and O. Dovgoshey, On statistical convergence of metric valued sequences, arXiv:1203.2584 [math.FA] 12 Mar, 2012.
M. Küçükaslan and U. Deger, On statistical boundedness of metric valued sequences, Eur. J. Pure Appl. Math. 5 (2012), no. 2, 174--186.
I. J. Maddox, Statistical convergence in a locally convex space, Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 1, 141--145.
D. Maharam, Finitely additive measures on the integers, Sankhya Ser. A 38 (1976), no. 1, 44--59.
G. D. Maio, L. D. R. Kocinac Statistical convergence in topology, Topology Appl. 156 (2008), no. 1, 46--55.
H. I. Miller, A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1811--1819.
S. Pehlivan and M. A. Mamedov, Statistical cluster points and turnpike, Optimization 48 (2000), no. 1, 93--106.
G. M. Peterson, Regular Matrix, Tramformations, Mc. Graw-Hill, London-New York-Toronto-Sydney, 1966.
D. Rath and B. C. Tripathy, Matrix maps on sequence spaces associated with sets of integers, Indian Jour. Pure Appl. Math. 27 (1996), no. 2, 197--206.
I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959) 361--375.
H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloquium Mathematicum 2 (1951) 73--74.
B. C. Tripathy and M. Sen, On generalized statistically convergent sequences, Indian J. Pure Appl. Math. 32 (2001), no. 11, 1689--1694.
B. C. Tripathy, M. Sen and S. Nath, I-convergence in probabilistic n-normed space, Soft Comput. 16 (2012), 1021--1027, DOI 10.1007/s00500-011-0799-8.
B. C. Tripathy and M. Sen, Paranormed I-convergent double sequence spaces associated with multiplier sequences, Kyungpook Math. J. 54 (2014), no. 2, 321--332.
B. C. Tripathy, S. Mahanta, On I-acceleration convergence of sequences, J. Franklin Inst. 347 (2010), no. 3, 591--598.
B. C. Tripathy, B. Hazarika and B. Choudhary, Lacunary I-convergent sequences, Kyungpook Math. J. 52 (2012), no. 4, 473--482.
A. Zygmund, Trigonometric series, Cambridge UK, 2nd edition, 1979.