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.

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 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 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 Korovkin type approximation theorems proved viaweighted αβ-equistatistical convergence for bivariate functions

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6253-6266
Author(s):  
Hüseyin Aktuğlu ◽  
Halil Gezer
Keyword(s):  
Uniform Convergence ◽  
Statistical Convergence ◽  
Linear Operators ◽  
Extension Problem ◽  
Real Numbers ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Definition Of ◽  
Korovkin Type Approximation Theorems ◽  
Bivariate Functions

Statistical convergence was extended to weighted statistical convergence in [24], by using a sequence of real numbers sk, satisfying some conditions. Later, weighted statistical convergence was considered in [35] and [19] with modified conditions on sk. Weighted statistical convergence is an extension of statistical convergence in the sense that, for sk = 1, for all k, it reduces to statistical convergence. A definition of weighted ??-statistical convergence of order ?, considered in [25] does not have this property. To remove this extension problem the definition given in [25] needs some modifications. In this paper, we introduced the modified version of weighted ??-statistical convergence of order ?, which is an extension of ??-statistical convergence of order ?. Our definition, with sk = 1, for all k, reduces to ??-statistical convergence of order ?. Moreover, we use this definition of weighted ??-statistical convergence of order ?, to prove Korovkin type approximation theorems via, weighted ??-equistatistical convergence of order ? and weighted ??-statistical uniform convergence of order ?, for bivariate functions on [0,?) x [0,?). Also we prove Korovkin type approximation theorems via ??-equistatistical convergence of order ? and ??-statistical uniform convergence of order ?, for bivariate functions on [0,?) x [0,?). Some examples of positive linear operators are constructed to show that, our approximation results works, but its classical and statistical cases do not work. Finally, rates of weighted ??-equistatistical convergence of order ? is introduced and discussed.

scholarly journals Ideal relatively uniform convergence with Korovkin and Voronovskaya types approximation theorems

Filomat ◽  
2019 ◽  
Vol 33 (14) ◽  
pp. 4549-4560 ◽  
Author(s):  
S.A. Mohiuddine ◽  
Bipan Hazarika ◽  
Mohammed Alghamdi
Keyword(s):  
Uniform Convergence ◽  
Approximation Theorem ◽  
Linear Operators ◽  
Korovkin Theorem ◽  
Korovkin Type Approximation Theorem ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Convergence Of Sequences ◽  
Relative Uniform Convergence ◽  
Relatively Uniform Convergence

We introduce the notion of ideally relative uniform convergence of sequences of real valued functions. We then apply this notion to prove Korovkin-type approximation theorem, and then construct an illustrative example by taking (p,q)-Bernstein operators which proves that our Korovkin theorem is stronger than its classical version as well as statistical relative uniform convergence. The rate of ideal relatively uniform convergence of positive linear operators by means of modulus of continuity is calculated. Finally, the Voronovskaya-type approximation theorem is also investigated.

scholarly journals A Class of Integral Operators that Fix Exponential Functions

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Carlo Bardaro ◽  
Ilaria Mantellini ◽  
Gumrah Uysal ◽  
Basar Yilmaz
Keyword(s):  
General Class ◽  
Pointwise Convergence ◽  
Type Theorem ◽  
Integral Operators ◽  
Exponential Functions ◽  
Convergence Theorems ◽  
Korovkin Type Theorem ◽  
Type Formula

AbstractIn this paper we introduce a general class of integral operators that fix exponential functions, containing several recent modified operators of Gauss–Weierstrass, or Picard or moment type operators. Pointwise convergence theorems are studied, using a Korovkin-type theorem and a Voronovskaja-type formula is obtained.

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