Statistical Equal Convergence On Weighted Spaces (original) (raw)
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A-Statistical Equal Approximation on Two Dimensional Weighted Spaces
4th International Symposium on Innovative Approaches in Engineering and Natural Sciences Proceedings, 2019
Korovkin type approximation theorems have very important role in the approximation theory. Many mathematicians investigate and improve these type of approximation theorems for various operators defined on different spaces via several new convergence methods. The convergence of a sequence of positive linear operators defined on weighted space was first studied by Gadjiev [Theorems of Korovkin type, Math. Zametki 20(1976), 781-786]. Then, these results were improved by many authors for different type of convergence methods. Recently, some authors study Korovkin type theorems for two variables functions by means of single and double sequences on weighted spaces. In this paper, we prove a Korovkin type approximation theorem for the notion of statistical equal convergence for double sequences on two dimensional weighted spaces. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work. Also, we compute the rate of stati...
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The concept of weighted statistical convergence was introduced and studied by Karakaya and Chishti (2009) [7]. In this paper, we modify the definition of weighted statistical convergence and find its relationship with the concept of statistical summability ðN; p n Þ due to Moricz and Orhan (2004) [10]. We apply this new summability method to prove a Korovkin type approximation theorem by using the test functions 1; e Àx ; e À2x. We apply the classical Baskakov operator to construct an example in support of our result.
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The Scientific World Journal, 2013
Korovkin-type theorem which is one of the fundamental methods in approximation theory to describe uniform convergence of any sequence of positive linear operators is discussed on weighted spaces, 1 ≤ < ∞ for univariate and multivariate functions, respectively. Furthermore, we obtain these types of approximation theorems by means of A-summability which is a stronger convergence method than ordinary convergence.
Korovkin type theorems for weighted approximation
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We present some new results related with approximation of functions in weighted spaces. We show that the theory of approximation by linear operators in the spaces C * ρ [0, ∞) can be reduced the same theory for the space C[0, 1] and that there are not finite Korovkin families for the space C ρ [0, ∞).
Some weighted statistical convergence and associated Korovkin and Voronovskaya type theorems
Journal of Applied Mathematics and Computing, 2020
In this paper, we propose to investigate a new weighted statistical convergence by applying the Nörlund-Cesáro summability method. Based upon this definition, we prove some properties of statistically convergent sequences and a kind of the Korovkin type theorems. We also study the rate of the convergence for this kind of weighted statistical convergence and a Voronovskaya type theorem. Keywords Weighted statistical convergence • Sequence spaces • Korovkin type theorem • Rate of convergence • Voronovskaya type theorem Mathematics Subject Classification 40G15 • 41A36 • 46A35 • 46A45 |K n | n if the limit exists, where the vertical bars in |K n | indicate the number of elements in the enclosed B Naim L. Braha
Korovkin type approximation theorems for weighted \alpha \beta α β -statistical convergence
Bulletin of Mathematical Sciences, 2015
The concept of αβ-statistical convergence was introduced and studied by Aktuglu (Korovkin type approximation theorems proved via αβ-statistical convergence, J Comput Appl Math 259:174-181, 2014). In this work, we generalize the concept of αβ-statistical convergence and introduce the concept of weighted αβstatistical convergence of order γ , weighted αβ-summability of order γ , and strongly weighted αβ-summable sequences of order γ. We also establish some inclusion relation, and some related results for these new summability methods. Furthermore, we prove Korovkin type approximation theorems through weighted αβ-statistical convergence and apply the classical Bernstein operator to construct an example in support of our result. Keywords Korovkin type theorems • Weighted αβ-summability • Rate of the weighted αβ-statistical convergent • Positive linear operator Mathematics Subject Classification 41A10 • 41A25 • 41A36 • 40A30 • 40G15 Communicated by S. K. Jain.
Mathematical Methods in the Applied Sciences, 2017
In the study of sequence spaces, classical convergence has got innumerable applications where the convergence of a sequence requires that almost all elements are to satisfy the convergence condition, that is, all of the elements of the sequence need to be in an arbitrarily small neighborhood of the limit. However, such restriction is relaxed in statistical convergence, where the validity of the convergence condition is achieved only for a majority of elements. The notion of statistical convergence was introduced and studied by Fast 1 and Steinhaus. 2 Recently, statistical convergence has been a dynamic research area due basically to the fact that it is more general than the classical convergence and such a theory as well as its various applications are discussed in the study in the areas of (for example) Fourier analysis, number theory, and approximation theory. For more details, see the recent works. 3-14 Let N be the set of natural numbers, and let K ⊆ N. Also, let K n = {k ∶ k ≦ n and k ∈ K},
Axioms, 2022
Here, in this article, we introduce and systematically investigate the ideas of deferred weighted statistical Riemann integrability and statistical deferred weighted Riemann summability for sequences of functions. We begin by proving an inclusion theorem that establishes a relation between these two potentially useful concepts. We also state and prove two Korovkin-type approximation theorems involving algebraic test functions by using our proposed concepts and methodologies. Furthermore, in order to demonstrate the usefulness of our findings, we consider an illustrative example involving a sequence of positive linear operators in conjunction with the familiar Bernstein polynomials. Finally, in the concluding section, we propose some directions for future research on this topic, which are based upon the core concept of statistical Lebesgue-measurable sequences of functions.
Λ2-Weighted statistical convergence and Korovkin and Voronovskaya type theorems
Applied Mathematics and Computation, 2015
summability 2-Weighted statistical convergence and Korovkin type theorems Rate of convergence and Voronovskaya type theorems Positive linear operators Bounded and continuous functions and modulus of continuity Nonincreasing and nondecreasing functions a b s t r a c t In this paper, we propose to introduce a new 2-weighted statistical convergence. Based upon this definition, we prove some Korovkin type theorems. We also find the rate of the convergence for this kind of weighted statistical convergence and derive some Voronovskaya type theorems.
Equi-Statistical Relative Convergence and Korovkin-Type Approximation
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Classical approximation theory has started with the proof of Weierstrass approximation theorem and after that Korovkin [Linear operators and approximation theory, Hindustan Publ. Corp, Delhi, 1960] first established the necessary and sufficient conditions for uniform convergence of a sequence of positive linear operators to a function f. In classical Korovkin theorem, most of the classical operators tend to converge to the value of the function being approximated. Also, the attention of researchers has been attracted to the notion of statistical convergence because of the fact that it is stronger than the classical convergence method. Furthermore, the concept of equi-statistical convergence is more general than the statistical uniform convergence. In this work, we introduce our new convergence method named equi-statistical relative convergence to demonstrate a Korovkin type approximation theorems which were proven by earlier authors. Then, we present an example in support of our definition and result presented in this paper. Finally, we compute the rate of the convergence.