A numerical comparative study of generalized Bernstein-Kantorovich operators

Uğur Kadak; Faruk Özger

doi:10.3934/mfc.2021021

Article Contents

2021, Volume 4, Issue 4: 311-332. Doi: 10.3934/mfc.2021021

This issue Previous Article A note on convergence results for varying interval valued multisubmeasures Next Article

A numerical comparative study of generalized Bernstein-Kantorovich operators

Uğur Kadak^1, and
Faruk Özger^2, ,

1.
Department of Mathematics, Gazi University, Ankara-06100, Turkey, Yaşamkent Street, Bolu-14100, Turkey
2.
Department of Engineering Sciences, İzmir Katip Çelebi University, İzmir-35620, Turkey

^*Corresponding author
^*Corresponding author

Received: May 31, 2021

Revised: July 31, 2021

Early access: September 2021

Published: November 2021

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Abstract

In this paper, a new generalization of the Bernstein-Kantorovich type operators involving multiple shape parameters is introduced. Certain Voronovskaja and Grüss-Voronovskaya type approximation results, statistical convergence and statistical rate of convergence of proposed operators are obtained by means of a regular summability matrix. Some illustrative graphics that demonstrate the convergence behavior, accuracy and consistency of the operators are given via Maple algorithms. The proposed operators are comprehensively compared with classical Bernstein, Bernstein-Kantorovich and other new modifications of Bernstein operators such as $ \lambda $-Bernstein, $ \lambda $-Bernstein-Kantorovich, $ \alpha $-Bernstein and $ \alpha $-Bernstein-Kantorovich operators.

Keywords:

Mathematics Subject Classification: Primary: 41A10, 41A25; Secondary: 41A36, 26A16, 40C05.

Citation:

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Figure 1. Generalized Bernstein basis polynomial with $ p = 4 $ and $ p = 5 $

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Figure 2. Generalized Bernstein basis polynomial with $ p = 6 $ and $ p = 8 $

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Figure 3. Approximations of operators $ \mathcal{K}_{p}(\vartheta; z;\lambda) $ with different $ p $ values

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Figure 4. Errors of approximation with different $ p $ values

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Figure 5. $[![]!]

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Figure 6. $[![]!]

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Table 1. Comparison of certain Kantorovich operators via maximum errors of approximation for the function $ \vartheta_1(z) $

$ p $	BK [22]	$ \lambda $-BK [1]	$ \alpha $-BK [37]	G. B. K.
4	0.587665e-6	0.555805e-6	0.584478e-6	0.555805e-6
5	0.490495e-6	0.450425e-6	0.490010e-6	0.449155e-6
6	0.421873e-6	0.381607e-6	0.422809e-6	0.375689e-6
8	0.330590e-6	0.296430e-6	0.332454e-6	0.281839e-6
10	0.272233e-6	0.244693e-6	0.274122e-6	0.224854e-6
16	0.178427e-6	0.163579e-6	0.179697e-6	0.139245e-6
20	0.145198e-6	0.134763e-6	0.146148e-6	0.110875e-6
40	0.752695e-7	0.722972e-7	0.755868e-7	0.547628e-7

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Table 2. Comparison of various operators via maximum errors of approximation for the function $ \vartheta_2(z) $

$p$	Bernstein [3]	$\lambda$-B [6]	$\alpha$-B [7]	BK [22]	$\lambda$-BK [1]	$\alpha$-BK [37]	G. B. K.
4	0.82355	0.94247	0.92642	0.78600	0.78600	0.78028	0.78600
5	0.76350	0.84155	0.83258	0.67014	0.67014	0.66573	0.67014
8	0.58677	0.61784	0.62170	0.49110	0.51367	0.51911	0.46399
10	0.49794	0.51722	0.52434	0.45009	0.46447	0.47119	0.38486
20	0.35516	0.35857	0.36421	0.33501	0.33783	0.34315	0.30638
40	0.25152	0.25210	0.25468	0.24304	0.24354	0.24604	0.23436

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Table 3. Comparison of operators via maximum errors for the function $ \vartheta_3(z) $

$ p $	$ \left\| \mathcal{K}_p(\vartheta_3;z;\lambda_k)-\vartheta_3(z) \right\| $	$ \left\|\mathcal{K}_p(\vartheta_3;z;\lambda_s)-\vartheta_3(z) \right\| $	$ \left\|B_p(\vartheta_3(z)) -\vartheta_3(z)\right\| $
$ 10 $	0.595	0.585	0.639
$ 20 $	0.379	0.375	0.392
$ 30 $	0.275	0.275	0.282

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