On the Rate of Convergence for Modified Baskakov Operators

2004 ◽  
Vol 44 (1) ◽  
pp. 102-107 ◽  
Author(s):  
Z. Walczak
Keyword(s):  
Rate Of Convergence ◽  
Baskakov Operators

On Global Inverse Theorems of Szász and Baskakov Operators

1979 ◽  
Vol 31 (2) ◽  
pp. 255-263 ◽  
Author(s):  
Z. Ditzian
Keyword(s):  
Rate Of Convergence ◽  
Exponential Growth ◽  
Polynomial Growth ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Baskakov Operators ◽  
Inverse Theorems ◽  
Image Position ◽  
Necessary And Sufficient

The Szász and Baskakov approximation operators are given by1.11.2respectively. For continuous functions on [0, ∞) with exponential growth (i.e. ‖ƒ‖A ≡ supx\ƒ(x)e–Ax\ < M) the modulus of continuity is defined by1.3where ƒ ∈ Lip* (∝, A) for some 0 < ∝ ≦ 2 if w2(ƒ, δ, A) ≦ Mδ∝ for all δ < 1. We shall find a necessary and sufficient condition on the rate of convergence of An(ƒ, x) (representing Sn(ƒ, x) or Vn(ƒ, x)) to ƒ(x) for ƒ(x) ∈ Lip* (∝, A). In a recent paper of M. Becker [1] such conditions were found for functions of polynomial growth (where (1 + \x\N)−1 replaced e–Ax in the above). M. Becker explained the difficulties in treating functions of exponential growth.

Rate of Convergence of Modified Baskakov Operators

1997 ◽  
Vol 30 (2) ◽  
pp. 339-346
Author(s):  
Vijay Gupta ◽  
D. Kumar
Keyword(s):  
Rate Of Convergence ◽  
Baskakov Operators

scholarly journals On the Rate of Convergence by Generalized Baskakov Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Yi Gao ◽  
Wenshuai Wang ◽  
Shigang Yue
Keyword(s):  
Uniform Convergence ◽  
Rate Of Convergence ◽  
Pointwise Convergence ◽  
Point Of View ◽  
Baskakov Operators ◽  
Computational Point

We firstly construct generalized Baskakov operatorsVn,α,q(f;x)and their truncated sumBn,α,q(f;γn,x). Secondly, we study the pointwise convergence and the uniform convergence of the operatorsVn,α,q(f;x), respectively, and estimate that the rate of convergence by the operatorsVn,α,q(f;x)is1/nq/2. Finally, we study the convergence by the truncated operatorsBn,α,q(f;γn,x)and state that the finite truncated sumBn,α,q(f;γn,x)can replace the operatorsVn,α,q(f;x)in the computational point of view provided thatlimn→∞nγn=∞.

scholarly journals Stancu-variant of generalized Baskakov operators

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2625-2632 ◽  
Author(s):  
Nadeem Rao ◽  
Abdul Wafi
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Type Theorem ◽  
Modulus Of Smoothness ◽  
Order Of Approximation ◽  
Baskakov Operators ◽  
Direct Estimate ◽  
Voronovskaya Type Theorem

In the present paper, we introduce Stancu-variant of generalized Baskakov operators and study the rate of convergence using modulus of continuity, order of approximation for the derivative of function f . Direct estimate is proved using K-functional and Ditzian-Totik modulus of smoothness. In the last, we have proved Voronovskaya type theorem.

scholarly journals Better approximation by a Durrmeyer variant of $ \alpha- $Baskakov operators

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Purshottam Narain Agrawal ◽  
Jitendra Kumar Singh
Keyword(s):  
Rate Of Convergence ◽  
Maximal Function ◽  
Modulus Of Smoothness ◽  
Order Of Approximation ◽  
Test Functions ◽  
Approximation Properties ◽  
Baskakov Operators ◽  
Convergence Behaviour ◽  
Durrmeyer Variant ◽  
Lipschitz Type Maximal Function

<p style='text-indent:20px;'>The aim of this paper is to study some approximation properties of the Durrmeyer variant of <inline-formula><tex-math id="M2">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-Baskakov operators <inline-formula><tex-math id="M3">\begin{document}$ M_{n,\alpha} $\end{document}</tex-math></inline-formula> proposed by Aral and Erbay [<xref ref-type="bibr" rid="b3">3</xref>]. We study the error in the approximation by these operators in terms of the Lipschitz type maximal function and the order of approximation for these operators by means of the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja and Gr<inline-formula><tex-math id="M4">\begin{document}$ \ddot{u} $\end{document}</tex-math></inline-formula>ss Voronovskaja type theorems are also established. Next, we modify these operators in order to preserve the test functions <inline-formula><tex-math id="M5">\begin{document}$ e_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ e_2 $\end{document}</tex-math></inline-formula> and show that the modified operators give a better rate of convergence. Finally, we present some graphs to illustrate the convergence behaviour of the operators <inline-formula><tex-math id="M7">\begin{document}$ M_{n,\alpha} $\end{document}</tex-math></inline-formula> and show the comparison of its rate of approximation vis-a-vis the modified operators.</p>

scholarly journals Rate of convergence of bounded variation functions by a Bézier-Durrmeyer variant of the Baskakov operators

2004 ◽  
Vol 2004 (9) ◽  
pp. 459-468 ◽  
Author(s):  
Vijay Gupta ◽  
Ulrich Abel
Keyword(s):  
Rate Of Convergence ◽  
Bounded Variation ◽  
Functions Of Bounded Variation ◽  
Baskakov Operators ◽  
Bounded Variation Functions ◽  
Durrmeyer Variant

We consider a Bézier-Durrmeyer integral variant of the Baskakov operators and study the rate of convergence for functions of bounded variation.

Generalized q-Baskakov operators

2011 ◽  
Vol 61 (4) ◽  
Author(s):  
Ali Aral ◽  
Vijay Gupta
Keyword(s):  
Rate Of Convergence ◽  
Weighted Norm ◽  
Shape Preserving ◽  
Baskakov Operators ◽  
Shape Preserving Properties

AbstractIn the present paper we propose a generalization of the Baskakov operators, based on q integers. We also estimate the rate of convergence in the weighted norm. In the last section, we study some shape preserving properties and the property of monotonicity of q-Baskakov operators.

scholarly journals Baskakov-Kantorovich operators reproducing affine functions

2021 ◽  
Vol 66 (4) ◽  
pp. 739-756
Author(s):  
Jorge Bustamante ◽  
Keyword(s):  
Rate Of Convergence ◽  
Weighted Spaces ◽  
Weighted Norm ◽  
Baskakov Operators ◽  
Affine Functions ◽  
Upper Estimate ◽  
Kantorovich Operators

We present a new Kantorovich modi cation of Baskakov operators which reproduce a ne functions. We present an upper estimate for the rate of convergence of the new operators in polynomial weighted spaces and characterize all functions for which there is convergence in the weighted norm.

scholarly journals Rate of Convergence of Modified Baskakov-Durrmeyer Type Operators for Functions of Bounded Variation

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Prashantkumar Patel ◽  
Vishnu Narayan Mishra
Keyword(s):  
Weight Function ◽  
Rate Of Convergence ◽  
Bounded Variation ◽  
Basis Function ◽  
Functions Of Bounded Variation ◽  
Baskakov Operators ◽  
Durrmeyer Type Operators

We study a certain integral modification of well-known Baskakov operators with weight function of beta basis function. We establish rate of convergence for these operators for functions having derivative of bounded variation. Also, we discuss Stancu type generalization of these operators.

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