Statistical extension of bounded sequence space

Yıl 2021, Cilt: 70 Sayı: 1, 82 - 99, 30.06.2021

Maya Altınok , Mehmet Küçükaslan , Umutcan Kaya

https://doi.org/10.31801/cfsuasmas.736132

Cited By: 6

Öz

In this paper by using natural density real valued bounded sequence space $l_{\infty}$ is extented and statistical bounded sequence space $l_{\infty}^{st}$ is obtained. Besides the main properties of the space $l_{\infty}^{st}$, it is shown that $l_{\infty}^{st}$ is a Banach space with a norm produced with the help of density. Also, it is shown
that there is no matrix extension of the space $l_{\infty}$ that its bounded sequences space covers $l_{\infty}^{st}$. Finally, it is shown that the space $l_{\infty}$ is a non-porous subset of $l_{\infty}^{st}$.

Anahtar Kelimeler

Natural density, , statistical convergence, statistical sup norm

Proje Numarası

Bu bir projenin sonuçları değildir.

Kaynakça

• Altınok, M., Küçükaslan, M., A-statistical convergence and A-statistical monotonicity, Applied Mathematics E-Notes, 13 (2013), 249-260.
• Altınok, M., Küçükaslan, M., Ideal limit superior-inferior, Gazi University Journal of Science, 30 (1) (2017), 401-411.
• Altınok, M., Küçükaslan, M., A-statistical supremum-infimum and A-statistical convergence, Azerbaijan Journal of Mathematics, 4 (2) (2014), 31-42.
• Altınok, M., Porosity supremum-infimum and porosity convergence, Konuralp Journal of Mathematics, 6 (1) (2018), 163-170.
• Bilalov, B., Nazarova, T., On statistical convergence in metric spaces, Journal of Mathematics Research, 7 (1) (2015), 37-43.
• Bilalov, B., Nazarova, T., On statistical type convergence in uniform spaces, Bull. of the Iranian Math. Soc., 42 (4) (2016), 975-986.
• Cabrera, M. O., Rosalsky, A., Ünver, M., Volodin, A., On the concept of B-statistical uniform integrability of weighted sums of random variables and the law of large numbers with mean convergence in the statistical sense, TEST, (2020).
• Et, M., Sengül, H., Some Cesaro type summability spaces of order α and lacunary statistical convergence of order α, Filomat, 28 (8) (2014), 1593-1602.
• Erdös, P., Tenenboum, G., Sur les densities de certaines suites dentiers, Proc. London Math. Soc., 59 (1989), 417-438.
• Fast, H., Sur la convergence statistique, Colloquium Mathematicae, 2 (3-4) (1951), 241-244.
• Fridy, J. A., On statistical convergence, Analysis, 5 (1985), 301-313.
• Fridy, J. A., Khan, M. K., Tauberian theorems via statistical convergnece, J.Math. Anal.Appl., 228 (1998), 73-95.
• Gadjiev, A. D., Orhan, C., Some approximation theorems via statistical convergnece, Rocky Mountain J. Math., 32 (2002), 129-138.
• Kaya, E, Küçükaslan, M., Wagner, R., On statistical convergence and statistical monotonicity, Annales Univ. Sci. Budapest. Sect. Comp., 39 (2013), 257-270.
• Küçükaslan, M., Altınok, M., Statistical supremum-infimum and statistical convergence, The Aligarh Bulletin of Mathematics, 32 (1-2) (2013), 1-16.
• Küçükaslan, M., Deger U., Dovgoshey O., On the Statistical Convergence of Metric-Valued Sequences, Ukrainian Mathematical Journal, 66 (5) (2014), 712-720.
• Lindenstrauss, J., Preiss, D., Tišer, J., Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces, Princeton University Press, 41 William Street, Princeton, New Jersey, 2012.
• Lonetti, P., Limit points of subsequences, Topology and its Applications, 263 (2019), 221-229.
• Miller, H. I., A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc., 347 (1995), 1881-1919.
• Pratulananda, D., Savas, E., On Generalized Statistical and Ideal Convergence of Metric-Valued Sequences, Ukrainian Mathematical Journal, 68 (1) (2017), 1849-1859.
• Salat, T., On statistical convergent sequences of real numbers, Math. Slovaca, 30 (2) (1980), 139-150.
• Sanjoy Ghosal, K., Statistical convergence of a sequence of random variables and limits theorems, Applications of Mathematics, 58 (4) (2013), 423-437.
• Sandeep G., Bhardwaj V.K., On deferred f-statistical convergnece, Kyunpook Math.J., 58 (2018), 91-103.
• Steinhaus, H., Sur la convergene ordinaire et la convergence asymptotique, Colloquium Mathematicae, 2 (3-4) (1951), 73-74.
• Zygmund, A., Trigonometric series, vol II, Cambiridge Univ Press, 1979.