Quantitative Voronovskaya type theorems and GBS operators of Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution

Parveen Bawa; Neha Bhardwaj; P. N. Agrawal

doi:10.3934/mfc.2022003

Article Contents

2022, Volume 5, Issue 4: 269-293. Doi: 10.3934/mfc.2022003

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Quantitative Voronovskaya type theorems and GBS operators of Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution

1.
Department of Mathematics, Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida 201303, India
2.
Department of Mathematics, School of Basic Sciences and Research, Sharda University, Greater Noida 201310, India
3.
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India

^*Corresponding author: Neha Bhardwaj
^*Corresponding author: Neha Bhardwaj

Received: September 2021

Revised: December 2021

Early access: February 2022

Published: November 2022

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Abstract

The motivation behind the current paper is to elucidate the approximation properties of a Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution. We construct quantitative-Voronovskaya and Grüss-Voronovskaya type theorems and determine the convergence estimates of the above operators. We also contrive the statistical convergence and talk about the approximation degree of a bivariate extension of these operators by exhibiting the convergence rate in terms of the complete and partial moduli of continuity. We build GBS (Generalized Boolean Sum) operators allied with the bivariate operators and estimate their convergence rate using mixed modulus of smoothness and Lipschitz class of B$ \ddot{o} $gel continuous functions. We also evaluate the order of approximation of the GBS operators in the spaces of B-continuous (Bögel continuous) and B-differentiable (Bögel differentiable) functions. In addition, we depict the comparison between the rate of convergence of the proposed bivariate operators and the corresponding GBS operators for some functions by graphical illustrations using MATLAB software.

Keywords:

Lupaş-Stancu operators,
quantitative-Voronovskaya and Grüss-Voronovskaya type theorems,
GBS operators,
B-differential,
B-continuous,
Pólya distribution,
Lipschitz class.

Mathematics Subject Classification: 41A25, 41A35.

Citation:

Full Text(HTML)

Figure 1. Graph showing convergence of $ T_{10,10,\frac{1}{10},\frac{1}{10}}^{\left ( \alpha ,\beta \right )} $ and $ S_{10,10\frac{1}{10},\frac{1}{10}}^{\left ( \alpha ,\beta \right )} $ to the function $ f\left ( x,r \right ) = x ^{4}+x ^{3}r+x r^{3}+r^{4} $

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Figure 2. Graph showing convergence of $ T_{50,50,\frac{1}{50},\frac{1}{50}}^{\left ( \alpha ,\beta \right )} $ and $ S_{50,50\frac{1}{50},\frac{1}{50}}^{\left ( \alpha ,\beta \right )} $ to the function $ f\left ( x,r \right ) = x ^{4}+x ^{3}r+x r^{3}+r^{4} $

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Figure 3. Graph showing convergence of $ T_{20,20,\frac{1}{20},\frac{1}{20}}^{\left ( \alpha ,\beta \right )} $ and $ S_{20,20\frac{1}{20},\frac{1}{20}}^{\left ( \alpha ,\beta \right )} $ to the function $ f\left ( x,r \right ) = \cos \left ( \frac{5x ^{3}}{x ^{2}+r^{2}} \right ) $

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Figure 4. Graph showing convergence of $ T_{50,50,\frac{1}{50},\frac{1}{50}}^{\left ( \alpha ,\beta \right )} $ and $ S_{50,50\frac{1}{50},\frac{1}{50}}^{\left ( \alpha ,\beta \right )} $ to the function $ f\left ( x,r \right ) = \cos \left ( \frac{5x ^{3}}{x ^{2}+r^{2}} \right ) $

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Table 1. Absolute error of operators $ T_{n ,m,\frac{1}{n },\frac{1}{m}}^{\left ( \alpha ,\beta \right )} $ and $ S_{n ,m,\frac{1}{n },\frac{1}{m}}^{\left ( \alpha ,\beta \right )} $ with function $ f\left ( x,r \right ) = \cos \left ( \frac{5x ^{3}}{x ^{2}+r^{2}} \right ) $ for $ m = n = 50 $ and $ \alpha_{1} = \alpha_{2} = \beta_{1} = \beta_{2} = 5 $

$ (x,r) $	$ \left\| T_{n ,m,\frac{1}{n },\frac{1}{m}}^{\left ( \alpha ,\beta \right )}-f\left ( x,r \right )\right\| $	$ \left\| S_{n ,m,\frac{1}{n },\frac{1}{m}}^{\left ( \alpha ,\beta \right )}-f\left ( x,r \right )\right\| $
(0.13, 0.20)	.4362	.4406
(0.15, 0.33)	.5185	.5210
(0.16, 0.39)	.5481	.5505
(0.18, 0.29)	.5274	.5330
(0.33, 0.63)	.6050	.6104
(0.37, 0.81)	.6365	.6404
(0.44, 0.37)	.1642	.1714
(0.83, 0.55)	.1761	.1827
(0.90, 1)	.6696	.6732
(1, 0.83)	.6696	.6732

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