scholarly journals Better degree of approximation by modified Bernstein-Durrmeyer type operators

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Purshottam Narain Agrawal ◽  
Şule Yüksel Güngör ◽  
Abhishek Kumar
Keyword(s):  
Present Article ◽  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Bernstein Operators ◽  
Approximation Of Functions ◽  
Careful Choice ◽  
Durrmeyer Type Operators ◽  
Durrmeyer Variant ◽  
Derivatives Of ◽  
Infinitely Differentiable Function

<p style='text-indent:20px;'>In the present article we investigate a Durrmeyer variant of the generalized Bernstein-operators based on a function <inline-formula><tex-math id="M1">\begin{document}$ \tau(x), $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M2">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> is infinitely differentiable function on <inline-formula><tex-math id="M3">\begin{document}$ [0, 1], \; \tau(0) = 0, \tau(1) = 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \tau^{\prime }(x)&gt;0, \;\forall\;\; x\in[0, 1]. $\end{document}</tex-math></inline-formula> We study the degree of approximation by means of the modulus of continuity and the Ditzian-Totik modulus of smoothness. A Voronovskaja type asymptotic theorem and the approximation of functions with derivatives of bounded variation are also studied. By means of a numerical example, finally we illustrate the convergence of these operators to certain functions through graphs and show a careful choice of the function <inline-formula><tex-math id="M5">\begin{document}$ \tau(x) $\end{document}</tex-math></inline-formula> leads to a better approximation than the generalized Bernstein-Durrmeyer type operators considered by Kajla and Acar [<xref ref-type="bibr" rid="b11">11</xref>].</p>

scholarly journals A NEW KIND OF THE VARIANT OF THE MODIFIED BERNSTEIN-KANTOROVICH OPERATORS DEFINED BY ¨OZARSLAN AND DUMAN

2021 ◽  
Author(s):  
Abhishek Kumar
Keyword(s):  
Rate Of Convergence ◽  
Present Article ◽  
Bounded Variation ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Type Theorem ◽  
Careful Choice ◽  
Derivatives Of ◽  
Kantorovich Operators ◽  
Infinitely Differentiable Function

In the present article, we dene a new kind of the modified Bernstein-Kantorovich operators defined by ¨ Ozarslan (https://doi.org/10.1080/01630563.2015.1079219) i.e. we introduce a new function ς(x) in the modified Bernstein-Kantorovich operators defined by Ozarslan with the property ({) is an infinitely differentiable function on [0; 1]; ς(0) = 0; ς(1) = 1 and ς’(x) > 0 for all x∈ [0; 1]. We substantiate an approximation theorem by using of the Bohman-Korovkins type theorem and scrutinize the rate of convergence with the aid of modulus of continuity, Lipschitz type functions for the our operators and the rate of convergence of functions by means of derivatives of bounded variation are also studied. We study an approximation theorem with the help of Bohman-Korovkins type theorem in A-Statistical convergence. Lastly, by means of a numerical example, we illustrate the convergence of these operators to certain functions through graphs with the help of MATHEMATICA and show that a careful choice of the function ς(x) leads to a better approximation results as compared to the modified Bernstein-Kantorovich operators defined by Ozarslan (https://doi.org/10.1080/01630563.2015.1079219).

scholarly journals A genuine family of Bernstein-Durrmeyer type operators based on Polya basis functions

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2611-2623 ◽  
Author(s):  
Trapti Neer ◽  
P.N. Agrawal
Keyword(s):  
Approximation Theorem ◽  
Local Approximation ◽  
Modulus Of Smoothness ◽  
Basis Functions ◽  
Global Approximation ◽  
Approximation Of Functions ◽  
Moment Estimates ◽  
Durrmeyer Type Operators ◽  
Direct Results ◽  
Classical Modulus

In this paper, we construct a genuine family of Bernstein-Durrmeyer type operators based on Polya basis functions. We establish some moment estimates and the direct results which include global approximation theorem in terms of classical modulus of continuity, local approximation theorem in terms of the second order Ditizian-Totik modulus of smoothness, Voronovskaya-type asymptotic theorem and a quantitative estimate of the same type. Lastly, we study the approximation of functions having a derivative of bounded variation.

scholarly journals General family of exponential operators

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4043-4060
Author(s):  
Km. Lipi ◽  
Naokant Deo
Keyword(s):  
Rate Of Convergence ◽  
Error Estimation ◽  
Bounded Variation ◽  
Approximation Theorem ◽  
Modulus Of Smoothness ◽  
Lipschitz Class ◽  
Approximation Properties ◽  
Approximation Of Functions ◽  
Direct Approximation ◽  
Derivatives Of

In this article, we deal with the approximation properties of Ismail-May operators [16] based on a non-negative real parameter ?. We provide some graphs and error estimation table for a numerical example depicting the convergence of our proposed operators. We further define the B?zier variant of these operators and give a direct approximation theorem using Ditizan-Totik modulus of smoothness and a Voronovoskaya type asymptotic theorem. We also study the error in approximation of functions having derivatives of bounded variation. Lastly, we introduce the bivariate generalization of Ismail May operators and estimate its rate of convergence for functions of Lipschitz class.

scholarly journals Dunkl analogue of Szász-mirakjan operators of blending type

2018 ◽  
Vol 16 (1) ◽  
pp. 1344-1356 ◽  
Author(s):  
Sheetal Deshwal ◽  
P.N. Agrawal ◽  
Serkan Araci
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Integral Form ◽  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Approximation Properties ◽  
Korovkin Theorem ◽  
Dunkl Analogue ◽  
Derivatives Of ◽  
Class Of Functions

AbstractIn the present work, we construct a Dunkl generalization of the modified Szász-Mirakjan operators of integral form defined by Pǎltanea [1]. We study the approximation properties of these operators including weighted Korovkin theorem, the rate of convergence in terms of the modulus of continuity, second order modulus of continuity via Steklov-mean, the degree of approximation for Lipschitz class of functions and the weighted space. Furthermore, we obtain the rate of convergence of the considered operators with the aid of the unified Ditzian-Totik modulus of smoothness and for functions having derivatives of bounded variation.

scholarly journals Trigonometric approximation of functions $f(x,y)$ of generalized Lipschitz class by double Hausdorff matrix summability method

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Abhishek Mishra ◽  
Vishnu Narayan Mishra ◽  
M. Mursaleen
Keyword(s):  
Double Fourier Series ◽  
Degree Of Approximation ◽  
Trigonometric Approximation ◽  
Lipschitz Class ◽  
Gamma Delta ◽  
Approximation Of Functions ◽  
Matrix Summability ◽  
Alpha Beta ◽  
Generalized Lipschitz Class ◽  
Hausdorff Matrix

AbstractIn this paper, we establish a new estimate for the degree of approximation of functions $f(x,y)$ f ( x , y ) belonging to the generalized Lipschitz class $Lip ((\xi _{1}, \xi _{2} );r )$ L i p ( ( ξ 1 , ξ 2 ) ; r ) , $r \geq 1$ r ≥ 1 , by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from $Lip ((\alpha ,\beta );r )$ L i p ( ( α , β ) ; r ) and $Lip(\alpha ,\beta )$ L i p ( α , β ) in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and $(C, \gamma , \delta )$ ( C , γ , δ ) means.

Some negative results for the interpolation monotone approximation of functions having a fractional derivative

2020 ◽  
pp. 122-127
Author(s):  
T. O. Petrova ◽  
I. P. Chulakov
Keyword(s):  
Fractional Derivative ◽  
Convex Function ◽  
Nondecreasing Function ◽  
Degree Of Approximation ◽  
Approximation Of Functions ◽  
Monotone Approximation ◽  
Convex Approximation ◽  
Negative Results ◽  
Positive Functions ◽  
Approximation Of A Function

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function $f є W^r [0,1]$ by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function $f є C^r [0,1] \cap \Delta^0$ where $\Delta^0$ is the set of positive functions on [0,1]. Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider $r є , r > 2$. In [8] is consider $r є , r > 2$. It was proved that for monotone approximation estimates of the form (1) are fails for $r є , r > 2$. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6]). In [5] is consider $r є , r > 2$. In [6] is consider $r є , r > 2$. It was proved that for convex approximation estimates of the form (1) are fails for $r є , r > 2$. In this paper the question of approximation of function $f є W^r \cap \Delta^1, r є (3,4)$ by algebraic polynomial $p_n є \Pi_n \cap \Delta^1$ is consider. The main result of the work generalize the result of work [8] for $r є (3,4)$.

scholarly journals Construction of Stancu-Type Bernstein Operators Based on Bézier Bases with Shape Parameter λ

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 316 ◽  
Author(s):  
Hari Srivastava ◽  
Faruk Özger ◽  
S. Mohiuddine
Keyword(s):  
Uniform Convergence ◽  
Shape Parameter ◽  
Approximation Theorem ◽  
Modulus Of Smoothness ◽  
Bernstein Operators ◽  
Global Approximation ◽  
Approximation Result ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Direct Approximation

We construct Stancu-type Bernstein operators based on Bézier bases with shape parameter λ ∈ [ - 1 , 1 ] and calculate their moments. The uniform convergence of the operator and global approximation result by means of Ditzian-Totik modulus of smoothness are established. Also, we establish the direct approximation theorem with the help of second order modulus of smoothness, calculate the rate of convergence via Lipschitz-type function, and discuss the Voronovskaja-type approximation theorems. Finally, in the last section, we construct the bivariate case of Stancu-type λ -Bernstein operators and study their approximation behaviors.

scholarly journals Drive for the divine

2015 ◽  
Vol 71 (3) ◽  
Author(s):  
Darryl Wooldridge
Keyword(s):  
Present Article ◽  
Imago Dei ◽  
Human Being ◽  
Human Beings ◽  
Relationship With God ◽  
Final Word ◽  
Current Article ◽  
Human Desire ◽  
Derivatives Of

Although the present article stands alone, it is a continuation of ‘Living in the not-yet’ (published in vol. 71, issue 1 of HTS). Both articles are derivatives of a larger study that discusses God as the centre of an often inarticulate and inchoate but innate human desire and pursuit to enjoy and reflect the divine image (imago Dei) in which every human being was created. The current article sets forth foundational considerations and speaks to the ineffaceable drive within humans to find God. It is a reciprocated drive – a response to God who first sought and continues to seek humans – a correlate and concomitant seeking in response to God. Although surely not the final word, this article discusses God as spirit and spiritual, by whom human beings have been created as imago Dei or God’s self-address, showing God’s heart as toward his creation, and humans most especially. Also discussed here is that humans are destined to join the perichoretic relationship that God has enjoyed from eternity. Moreover, in his ascension and glory, Jesus sends the Spirit of adoption into creation so that human creation might enter this same perichoretic relationship with God.

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