scholarly journals A Certain Class of Relatively Equi-Statistical Fuzzy Approximation Theorems

2020 ◽  
Vol 13 (5) ◽  
pp. 1212-1230
Author(s):  
Susanta Kumar Paikray ◽  
Priyadarsini Parida ◽  
S. A. Mohiuddine
Keyword(s):  
Statistical Convergence ◽  
Type Theorem ◽  
Point Of View ◽  
Scale Function ◽  
Korovkin Type Theorem ◽  
Approximation Theorems ◽  
Fuzzy Approximation ◽  
Inclusion Relations ◽  
Korovkin Type Theorems ◽  
Sequence Of Functions

The aim of this paper is to introduce the notions of relatively deferred Nörlund uniform statistical convergence as well as relatively deferred Norlund point-wise statistical convergence through the dierence operator of fractional order of fuzzy-number-valued sequence of functions, and a type of convergence which lies between aforesaid notions, namely, relatively deferred Nörlund equi-statistical convergence. Also, we investigate the inclusion relations among these aforesaidnotions. As an application point of view, we establish a fuzzy approximation (Korovkin-type) theorem by using our new notion of relatively deferred Norlund equi-statistical convergence and intimate that this result is a non-trivial generalization of several well-established fuzzy Korovkin-type theorems which were presented in earlier works. Moreover, we estimate the fuzzy rate of the relatively deferred Nörlund equi-statistical convergence involving a non-zero scale function by using the fuzzy modulus of continuity.

scholarly journals Korovkin-Type Theorems in WeightedLp-Spaces via Summation Process

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Tuncer Acar ◽  
Fadime Dirik
Keyword(s):  
Uniform Convergence ◽  
Approximation Theory ◽  
Type Theorem ◽  
Linear Operators ◽  
Multivariate Functions ◽  
Korovkin Type Theorem ◽  
Approximation Theorems ◽  
Convergence Method ◽  
Summation Process ◽  
Korovkin Type Theorems

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 weightedLpspaces,1≤p<∞for univariate and multivariate functions, respectively. Furthermore, we obtain these types of approximation theorems by means of𝒜-summability which is a stronger convergence method than ordinary convergence.

Some properties of modified Szász–Mirakyan operators in polynomial spaces via the power summability method

2020 ◽  
Vol 26 (1) ◽  
pp. 79-90 ◽  
Author(s):  
Naim L. Braha
Keyword(s):  
Statistical Convergence ◽  
Type Theorem ◽  
Summability Method ◽  
Korovkin Type Theorem ◽  
Summability Methods ◽  
Voronovskaya Type Theorem

AbstractIn this paper we will prove the Korovkin type theorem for modified Szász–Mirakyan operators via A-statistical convergence and the power summability method. Also we give the rate of the convergence related to the above summability methods, and in the last section, we give a kind of Voronovskaya type theorem for A-statistical convergence and Grüss–Voronovskaya type theorem.

scholarly journals Korovkin-type theorems on $$B({\mathcal {H}})$$ and their applications to function spaces

2021 ◽  
Author(s):  
Wolfram Bauer ◽  
V. B. Kiran Kumar ◽  
Rahul Rajan
Keyword(s):  
Toeplitz Operators ◽  
Recent Literature ◽  
Fock Space ◽  
Banach Algebras ◽  
Type Theorem ◽  
Function Spaces ◽  
Korovkin Type Theorem ◽  
Type Approximation ◽  
Korovkin Type Theorems ◽  
Toeplitz Structure

AbstractWe prove Korovkin-type theorems in the setting of infinite dimensional Hilbert space operators. The classical Korovkin theorem unified several approximation processes. Also, the non-commutative versions of the theorem were obtained in various settings such as Banach algebras, $$C^{*}$$ C ∗ -algebras and lattices etc. The Korovkin-type theorem in the context of preconditioning large linear systems with Toeplitz structure can be found in the recent literature. In this article, we obtain a Korovkin-type theorem on $$B({\mathcal {H}})$$ B ( H ) which generalizes all such results in the recent literature. As an application of this result, we obtain Korovkin-type approximation for Toeplitz operators acting on various function spaces including Bergman space $$A^{2}({\mathbb {D}})$$ A 2 ( D ) , Fock space $$F^{2}({\mathbb {C}})$$ F 2 ( C ) etc. These results are closely related to the preconditioning problem for operator equations with Toeplitz structure on the unit disk $${\mathbb {D}}$$ D and on the whole complex plane $${\mathbb {C}}$$ C . It is worthwhile to notice that so far such results are available for Toeplitz operators on circle only. This also establishes the role of Korovkin-type approximation techniques on function spaces with certain oscillation property. To address the function theoretic questions using these operator theory tools will be an interesting area of further research.

Korovkin-type theorems and applications

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Nazim Mahmudov
Keyword(s):  
Type Theorem ◽  
Linear Operators ◽  
Bernstein Operator ◽  
Approximation Properties ◽  
Korovkin Type Theorem ◽  
Quantitative Results ◽  
Korovkin Type Theorems

AbstractLet {T n} be a sequence of linear operators on C[0,1], satisfying that {T n (e i)} converge in C[0,1] (not necessarily to e i) for i = 0,1,2, where e i = t i. We prove Korovkin-type theorem and give quantitative results on C 2[0,1] and C[0,1] for such sequences. Furthermore, we define King’s type q-Bernstein operator and give quantitative results for the approximation properties of such operators.

scholarly journals Statistically and Relatively Modular Deferred-Weighted Summability and Korovkin-Type Approximation Theorems

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 448 ◽  
Author(s):  
H. Srivastava ◽  
B. Jena ◽  
S. Paikray ◽  
U. Misra
Keyword(s):  
Statistical Convergence ◽  
Approximation Theorem ◽  
Double Sequence ◽  
Modular Space ◽  
Double Sequences ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Korovkin Type Approximation Theorems ◽  
Kantorovich Operators ◽  
Sequence Of Functions

The concept of statistically deferred-weighted summability was recently studied by Srivastava et al. . The present work is concerned with the deferred-weighted summability mean in various aspects defined over a modular space associated with a generalized double sequence of functions. In fact, herein we introduce the idea of relatively modular deferred-weighted statistical convergence and statistically as well as relatively modular deferred-weighted summability for a double sequence of functions. With these concepts and notions in view, we establish a theorem presenting a connection between them. Moreover, based upon our methods, we prove an approximation theorem of the Korovkin type for a double sequence of functions on a modular space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results. Finally, an illustrative example is provided here by the generalized bivariate Bernstein–Kantorovich operators of double sequences of functions in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.

scholarly journals Generalized weighted statistical convergence of double sequences and applications

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 753-762 ◽  
Author(s):  
Muhammed Cinar ◽  
Mikail Et
Keyword(s):  
Statistical Convergence ◽  
Type Theorem ◽  
Summability Method ◽  
Double Sequences ◽  
Korovkin Type Theorem

In this paper we introduce the concept generalized weighted statistical convergence of double sequences. Some relations between weighted (?,?)-statistical convergence and strong (N???,p,q,?,?)-summablity of double sequences are examined. Furthemore, we apply our new summability method to prove a Korovkin type theorem.

Statistical Korovkin-type theory for matrix-valued functions

2011 ◽  
Vol 48 (4) ◽  
pp. 489-508
Author(s):  
Oktay Duman ◽  
Esra Erkuş-Duman
Keyword(s):  
Type Theory ◽  
Statistical Convergence ◽  
Toeplitz Matrices ◽  
Type Theorem ◽  
Positive Operators ◽  
Korovkin Type Theorem ◽  
Linear Positive Operators ◽  
Numer Anal ◽  
Matrix Spaces

In this paper, using the notion of A-statistical convergence from the summability theory, we obtain a Korovkin-type theorem for the approximation by means of matrixvalued linear positive operators. We show that our theorem is more applicable than the result introduced by S. Serra-Capizzano [A Korovkin based approximation of multilevel Toeplitz matrices (with rectangular unstructured blocks) via multilevel trigonometric matrix spaces, SIAM J. Numer. Anal., 36 (1999), 1831–1857]. Furthermore, we compute the A-statistical rates of our approximation.

scholarly journals Some Properties of Kantorovich-Stancu-Type Generalization of Szász Operators including Brenke-Type Polynomials via Power Series Summability Method

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Naim Latif Braha ◽  
Toufik Mansour ◽  
Mohammad Mursaleen
Keyword(s):  
Power Series ◽  
Statistical Convergence ◽  
Type Theorem ◽  
Summability Method ◽  
Korovkin Type Theorem ◽  
Szász Operators

In this paper, we study the Kantorovich-Stancu-type generalization of Szász-Mirakyan operators including Brenke-type polynomials and prove a Korovkin-type theorem via the T-statistical convergence and power series summability method. Moreover, we determine the rate of the convergence. Furthermore, we establish the Voronovskaya- and Grüss-Voronovskaya-type theorems for T-statistical convergence.

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