Chlodowsky type $\left( \lambda,q\right)$-Bernstein Stancu operators of Pascal rough triple sequences

Document Type : Research Paper

Authors

1 Department of Basic Eng.Sci. (Math.Sect.) Malatya Turgut Ozal University, Malatya, Turkey

2 Department of Mathematics, SASTRA University, Thanjavur-613 401, India

3 Department of Mathematics, Inonu University, Malatya, Turkey

Abstract

The fundamental concept of statistical convergence first was put forward by Steinhaus and at the same time but also  by Fast \cite{Fast} independently both for complex and real sequences. In fact, the convergence in terms of statistical     manner can be seen as a generalized form of the common convergence notion that is in the parallel of the theory of usual convergence. Measuring how large a subset of the set of natural number can be possible by means of asymptotic    density. It is intuitively known that positive integers are in fact far beyond the fact that they are perfect squares. This is due to the fact that each perfect square is positive and besides at the same time there are many other positive integers. But it is also known that the set consisting of integers which are positive is not larger than that of those which are perfect squares: both of those sets are countable and infinite and therefore can be considered in terms of $1$-to-$1$ correspondence. However, when the natural numbers are scanned for increasing order, then the squares are seen     increasingly scarcity. It is at this point that the concept of natural density comes into out help and this intuition becomes more precise. In this study, the above mentioned statistical convergence and asymptotic density concepts are     examined in a new space and an attempt is made to fill a gap in the literature as follows. Stancu type extension of the widely known Chlodowsky type \linebreak$\left( \lambda,q\right)  $-operators is going to be introduced. Moreover, the
    description of the novel rough statistical convergence having Pascal Fibonacci binomial matrix is going to be presented and several general characteristics of rough statistical convergence are taken into consideration. In the second place, the approximation theory is investigated as the rate of the rough statistical convergence of Chlodowsky type $\left(\lambda,q\right)$-operators.

Keywords


[1] S. Aytar, Rough statistical convergence, Numerical Functional Analysis and Optimization. An International Journal 29(3-4), 291{303 (2008).
[2] S. Debnath, B. Sarma, B. C. Das, Some generalized triple sequence spaces of real numbers, Journal of nonlinear analysis and optimization. Theory and Applications 6(1),71{78 (2015).
[3] A. J. Dutta, A. Esi, B. C. Tripathy, Statistically convergent triple sequence spaces de ned by Orlicz function, Journal of Mathematical Analysis 4(2), 16{22 (2013).
[4] A. Esi, On some triple almost lacunary sequence spaces de ned by Orlicz functions, Research and Reviews: Discrete Mathematical Structures 1(2), 16{25 (2014).
[5] A. Esi, S. Araci, M. Acikgoz, Statistical convergence of Bernstein operators, Applied Mathematics & Information Sciences 10(6), 2083{2086 (2016).
[6] A. Esi, S. Araci, A. Esi, -statistical convergence of Bernstein polynomial sequences, Advances and Applications in Mathematical Sciences 16(3), 113{119 (2017).
[7] A. Esi, M. N. Catalbas, Almost convergence of triple sequences, Global Journal of Mathematical Analysis 2(1), 6{10 (2014).
[8] A. Esi, E. Savas, On lacunary statistically convergent triple sequences in probabilistic normed space, Applied Mathematics & Information Sciences 9(5), 2529{2534 (2015).
[9] A. Esi, N. Subramanian, Generalized rough Cesaro and lacunary statistical Triple difference sequence spaces inprobability of fractional order de ned by Musielak Orlicz function, International Journal of Analysis and Applications 16(1), 16{24 (2018).
[10] A. Esi, N. Subramanian, On triple sequence spaces of Bernstein operator of 3 of rough 􀀀 statistical convergence in probability of random variables de ned by Musielak-Orlicz function, International Journal of Open Problems in Computer Science and Mathematics 11(2), 62{70 (2018).
[11] A. Esi, N. Subramanian, A. Esi, On triple sequence space of Bernstein operator of rough I􀀀 convergence Pre-Cauchy sequences, Proyecciones Journal of Mathematics 36(4) 567{587 (2017).
[12] A. Esi, N. Subramanian, V. A. Khan, The rough intuitionistic fuzzy Zweier lacunary ideal convergence of triple sequence spaces, Journal of Mathematics and Statistics 14(1),72{78 (2018).
[13] J. Fathi, R. Lashkaripour, On the  ne spectra of the generalized di erence operator uv over the sequence space c0, Journal of Mahani Mathematical Research 1(1), 1{12 (2012).
[14] H. Fast, Sur la convergence statistique, Colloquium Mathematicum 2, 241{244 (1951).
[15] B. Hazarika, N. Subramanian, A. Esi, On rough weighted ideal convergence of triple sequence of Bernstein polynomials, Proceedings of the Jangjeon Mathematical Society. Memoirs of the Jangjeon Mathematical Society 21(3), 497{506 (2018).
[16] S. K. Pal, D. Chandra, S. Dutta, Rough ideal convergence, Hacettepe Journal of Mathematics and Statistics, 42(6), 633{640 (2013).
[17] H. X. Phu, Rough convergence in normed linear spaces, Numerical Functional Analysis and Optimization. An International Journal 22(1-2), 199{222 (2001).
[18] A. Sahiner, M. Gurdal, F. K. Duden, Triple sequences and their statistical convergence, Selcuk Journal of Applied Mathematics 8(2), 49{55 (2007).
[19] A. Sahiner, B. C. Tripathy, Some I-related properties of triple sequences, Selcuk Journal of Applied Mathematics 9(2), 9{18 (2008).
[20] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloquium Mathematicum 2, 73{74 (1951).
[21] N. Subramanian, A. Esi, The generalized tripled di erence of 3 sequence spaces, Global Journal of Mathematical Analysis 3(2), 54{60 (2015).
[22] N. Subramanian, A. Esi, On triple sequence space of Bernstein operator of 3 of rough -statistical convergence in probability de nited by Musielak-Orlicz function p-metric, Electronic Journal of Mathematical Analysis and Applications 6(1), 198{203 (2018).
[23] N. Subramanian, A. Esi, M. K. Ozdemir, Rough statistical convergence on triple sequence of Bernstein operator of random variables in probability, Songklanakarin Journal of Science and Technology 41(3), 567{579 (2018).
[24] S. Velmurugan, N. Subramanian, Bernstein operator of rough -statistically and Cauchy sequences convergence on triple sequence spaces, Journal of Indian Mathematical Society 85(1{2), 257-265 (2018).