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 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.

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 Positive operators and approximation in function spaces on completely regular spaces

2003 ◽  
Vol 2003 (61) ◽  
pp. 3841-3871 ◽  
Author(s):  
Francesco Altomare ◽  
Sabrina Diomede
Keyword(s):  
Natural Extension ◽  
Function Spaces ◽  
Positive Operators ◽  
Linear Operators ◽  
Topological Spaces ◽  
Approximation Properties ◽  
Completely Regular ◽  
Borel Measures ◽  
Korovkin Type Theorems ◽  
Locally Convex

We discuss the approximation properties of nets of positive linear operators acting on function spaces defined on Hausdorff completely regular spaces. A particular attention is devoted to positive operators which are defined in terms of integrals with respect to a given family of Borel measures. We present several applications which, in particular, show the advantages of such a general approach. Among other things, some new Korovkin-type theorems on function spaces on arbitrary topological spaces are obtained. Finally, a natural extension of the so-called Bernstein-Schnabl operators for convex (not necessarily compact) subsets of a locally convex space is presented as well.

scholarly journals Convergence properties of Ibragimov-Gadjiev-Durrmeyer operators

2015 ◽  
Vol 24 (1) ◽  
pp. 17-26
Author(s):  
EMRE DENIZ ◽  
◽  
ALI ARAL ◽  
Keyword(s):  
Weighted Space ◽  
Type Theorem ◽  
Approximation Error ◽  
Weighted Norm ◽  
Approximation Properties ◽  
Convergence Properties ◽  
Korovkin Type Theorem ◽  
Durrmeyer Operators ◽  
Weighted Modulus ◽  
Direct Approximation

The purpose of the present paper is to study the local and global direct approximation properties of the Durrmeyer type generalization of Ibragimov Gadjiev operators defined in [Aral, A. and Acar, T., On Approximation Properties of Generalized Durrmeyer Operators, (submitted)]. The results obtained in this study consist of Korovkin type theorem which enables us to approximate a function uniformly by new Durrmeyer operators, and estimate for approximation error of the operators in terms of weighted modulus of continuity. These results are obtained for the functions which belong to weighted space with polynomial weighted norm by new operators which act on functions defined on the non compact interval [0.∞). We finally present a direct approximation result.

scholarly journals A Note on New Construction of Meyer-König and Zeller Operators and Its Approximation Properties

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3275
Author(s):  
Qing-Bo Cai ◽  
Khursheed J. Ansari ◽  
Fuat Usta
Keyword(s):  
Functional Analysis ◽  
Local Approximation ◽  
Linear Operators ◽  
Test Functions ◽  
Approximation Properties ◽  
Asymptotic Type ◽  
New Construction ◽  
Quantitative Results ◽  
Theoretical Results ◽  
Significant Point

The topic of approximation with positive linear operators in contemporary functional analysis and theory of functions has emerged in the last century. One of these operators is Meyer–König and Zeller operators and in this study a generalization of Meyer–König and Zeller type operators based on a function τ by using two sequences of functions will be presented. The most significant point is that the newly introduced operator preserves {1,τ,τ2} instead of classical Korovkin test functions. Then asymptotic type formula, quantitative results, and local approximation properties of the introduced operators are given. Finally a numerical example performed by MATLAB is given to visualize the provided theoretical results.

scholarly journals A Korovkin type theorem for weighted spaces of continuous functions

1997 ◽  
Vol 55 (2) ◽  
pp. 239-248 ◽  
Author(s):  
Walter Roth
Keyword(s):  
Type Theorem ◽  
Weighted Approximation ◽  
Continuous Functions ◽  
Linear Operators ◽  
Weighted Spaces ◽  
Korovkin Type Theorem ◽  
Korovkin Type Approximation Theorem ◽  
Spaces Of Continuous Functions ◽  
Locally Convex Topologies ◽  
Locally Convex

We prove a Korovkin type approximation theorem for positive linear operators on weighted spaces of continuous real-valued functions on a compact Hausdorff space X. These spaces comprise a variety of subspaces of C (X) with suitable locally convex topologies and were introduced by Nachbin 1967 and Prolla 1977. Some early Korovkin type results on the weighted approximation of real-valued functions in one and several variables with a single weight function are due to Gadzhiev 1976 and 1980.

scholarly journals Quantitative estimates for the Lupaş q-analogue of the Bernstein operator

2011 ◽  
Vol 44 (1) ◽  
Author(s):  
Zoltán Finta
Keyword(s):  
Modulus Of Smoothness ◽  
Bernstein Operator ◽  
Approximation Properties ◽  
Global Approximation ◽  
Approximation Theorems ◽  
Quantitative Estimates ◽  
Quantitative Results

AbstractWe establish quantitative results for the approximation properties of the q-analogue of the Bernstein operator defined by Lupaş in 1987 and for the approximation properties of the limit Lupaş operator introduced by Ostrovska in 2006, via Ditzian-Totik modulus of smoothness. Our results are local and global approximation theorems.

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.

scholarly journals APPROXIMATION PROPERTIES OF MODIFIED BASKAKOV GAMMA OPERATORS

2021 ◽  
pp. 125
Author(s):  
Seda Arpagus ◽  
Ali Olgun
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Type Theorem ◽  
Weighted Spaces ◽  
Approximation Properties ◽  
Korovkin Type Theorem ◽  
Gamma Operator ◽  
Voronovskaya Type Theorem

In this present paper, we study an approximation properties of modified Baskakov-Gamma operator. Using Korovkin type theorem we first give approximation properties of this operator. Secondly, we compute the rate of convergence of this operator by means of the modulus of continuity and we give an approximation properties of weighted spaces. Finally, we study the Voronovskaya type theorem of this operator.

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