Density by moduli and statistical boundedness.(English)Zbl 1470.40014

Summary: We have generalized the notion of statistical boundedness by introducing the concept of $$f$$-statistical boundedness for scalar sequences where $$f$$ is an unbounded modulus. It is shown that bounded sequences are precisely those sequences which are $$f$$-statistically bounded for every unbounded modulus $$f$$. A decomposition theorem for $$f$$-statistical convergence for vector valued sequences and a structure theorem for $$f$$-statistical boundedness have also been established.

MSC:

 40A35 Ideal and statistical convergence 46A45 Sequence spaces (including Köthe sequence spaces)
Full Text:

References:

 [1] Zygmund, A., Trigonometric Series, (1979), Cambridge, UK: Cambridge University Press, Cambridge, UK · JFM 58.0296.09 [2] Steinhaus, H., Sur la convergence ordinaire et la convergence asymptotique, Colloquium Mathematicum, 2, 73-74, (1951) [3] Fast, H., Sur la convergence statistique, Colloquium Mathematicum, 2, 241-244, (1951) · Zbl 0044.33605 [4] Schoenberg, I. J., The integrability of certain functions and related summability methods, The American Mathematical Monthly, 66, 361-375, (1959) · Zbl 0089.04002 [5] Aizpuru, A.; Listán-García, M. C.; Rambla-Barreno, F., Double density by moduli and statistical convergence, Bulletin of the Belgian Mathematical Society, 19, 4, 663-673, (2012) · Zbl 1264.40005 [6] Aizpuru, A.; Listán-García, M. C.; Rambla-Barreno, F., Density by moduli and statistical convergence, Quaestiones Mathematicae, 1-6, (2014) · Zbl 1426.40002 [7] Bhardwaj, V. K.; Bala, I., On weak statistical convergence, International Journal of Mathematics and Mathematical Sciences, 2007, (2007) · Zbl 1151.46013 [8] Bhardwaj, V. K.; Bala, I., On lacunary generalized difference sequence spaces defined by Orlicz functions in a seminormed space and $$\Delta_q^m$$ mlacunary statistical convergence, Demonstratio Mathematica, 41, 2, 415-424, (2008) · Zbl 1156.40001 [9] Bhardwaj, V. K.; Dhawan, S., f-statistical convergence of order α and strong Cesàro summability of order α with respect to a modulus, Journal of Inequalities and Applications, 2015, article 332, (2015) · Zbl 1351.40004 [10] Bhardwaj, V. K.; Gupta, S., On some generalizations of statistical boundedness, Journal of Inequalities and Applications, 2014, article 12, (2014) · Zbl 1327.40002 [11] Bhardwaj, V. K.; Gupta, S.; Mohiuddine, S. A.; Kılıçman, A., On lacunary statistical boundedness, Journal of Inequalities and Applications, 2014, article 311, (2014) · Zbl 1339.40006 [12] Et, M.; Çinar, M.; Karakaş, M., On λ-statistical convergence of order α of sequences of function, Journal of Inequalities and Applications, 2013, article 204, (2013) · Zbl 1286.40002 [13] Karakus, S., Statistical convergence on probabilistic normed spaces, Mathematical Communications, 12, 1, 11-23, (2007) · Zbl 1147.54016 [14] Karakus, S.; Demirci, K.; Duman, O., Statistical convergence on intuitionistic fuzzy normed spaces, Chaos, Solitons and Fractals, 35, 4, 763-769, (2008) · Zbl 1139.54006 [15] Savas, E.; Mohiuddine, S. A., $$\lambda$$ Statistically convergent double sequences in probabilistic normed spaces, Mathematica Slovaca, 62, 1, 99-108, (2012) · Zbl 1274.40016 [16] Niven, I.; Zuckerman, H. S., An Introduction to the Theory of Numbers, (1980), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0431.10001 [17] Burgin, M.; Duman, O., Statistical convergence and convergence in statistics [18] Fridy, J. A., On statistical convergence, Analysis, 5, 4, 301-313, (1985) · Zbl 0588.40001 [19] Fridy, J. A.; Orhan, C., Statistical limit superior and limit inferior, Proceedings of the American Mathematical Society, 125, 12, 3625-3631, (1997) · Zbl 0883.40003 [20] Tripathy, B. C., On statistically convergent and statistically bounded sequences, Bulletin of the Malaysian Mathematical Sciences Society—Second Series, 20, 1, 31-33, (1997) · Zbl 0926.40001 [21] Nakano, H., Concave modulars, Journal of the Mathematical Society of Japan, 5, 29-49, (1953) · Zbl 0050.33402 [22] Ruckle, W. H., FK spaces in which the sequence of coordinate vectors is bounded, Canadian Journal of Mathematics, 25, 973-978, (1973) · Zbl 0267.46008 [23] Maddox, I. J., Sequence spaces defined by modulus, Mathematical Proceedings of the Cambridge Philosophical Society, 101, 523-527, (1987) · Zbl 0631.46009 [24] Connor, J., On strong matrix summability with respect to a modulus and statistical convergence, Canadian Mathematical Bulletin, 32, 2, 194-198, (1989) · Zbl 0693.40007 [25] Pehlivan, S., Strongly almost convergent sequences defined by a modulus and uniformly statistical convergence, Soochow Journal of Mathematics, 20, 2, 205-211, (1994) · Zbl 0808.40007 [26] Pehlivan, S.; Fisher, B., Some sequence spaces defined by a modulus, Mathematica Slovaca, 45, 3, 275-280, (1995) · Zbl 0852.40002 [27] Kolk, E., F-seminormed sequence spaces defined by a sequence of modulus functions and strong summability, Indian Journal of Pure and Applied Mathematics, 28, 11, 1547-1566, (1997) · Zbl 0920.46002 [28] Ghosh, D.; Srivastava, P. D., On some vector valued sequence spaces defined using a modulus function, Indian Journal of Pure and Applied Mathematics, 30, 8, 819-826, (1999) · Zbl 0935.46008 [29] Bhardwaj, V. K.; Singh, N., Some sequence spaces defined by $$\overset{-}{N}, p_n$$, pnummability and a modulus function, Indian Journal of Pure and Applied Mathematics, 32, 12, 1789-1801, (2001) · Zbl 0991.40003 [30] Bhardwaj, V. K.; Singh, N., Banach space valued sequence spaces defined by a modulus, Indian Journal of Pure and Applied Mathematics, 32, 12, 1869-1882, (2001) · Zbl 1021.46020 [31] Çolak, R., Lacunary strong convergence of difference sequence spaces with respect to a modulus function, Filomat, 17, 9-14, (2003) · Zbl 1049.40002 [32] Altin, Y.; Et, M., Generalized difference sequence spaces defined by a modulus function in a locally convex space, Soochow Journal of Mathematics, 31, 2, 233-243, (2005) · Zbl 1085.46501 [33] Bhardwaj, V. K.; Dhawan, S., Density by moduli and lacunary statistical convergence, Abstract and Applied Analysis · Zbl 1376.40002 [34] Connor, J. S., The statistical and strong p-Cesàro convergence of sequences, Analysis, 8, 1-2, 47-63, (1988) · Zbl 0653.40001 [35] Šalát, T., On statistically convergent sequences of real numbers, Mathematica Slovaca, 30, 2, 139-150, (1980) · Zbl 0437.40003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.