Some new inequalities and numerical results of bivariate Bernstein-type operator including Bézier basis and its GBS operator

Esma Yıldız Özkan; Nesibe Nur Akpınar

doi:10.3934/mfc.2022045

Article Contents

2023, Volume 6, Issue 3: 500-511. Doi: 10.3934/mfc.2022045

This issue Previous Article Degree of convergence of a function in generalized Zygmund space Next Article On approximation of unbounded functions by certain modified Bernstein operators

Some new inequalities and numerical results of bivariate Bernstein-type operator including Bézier basis and its GBS operator

Esma Yıldız Özkan^1, , and
Nesibe Nur Akpınar^2,

1.
Department of Mathematics, Faculty of Sciences, Gazi University, 06500 Ankara, Turkey
2.
Department of Mathematics, Graduate School of Natural and Applied Sciences, Gazi University, 06500 Ankara, Turkey

^*Corresponding author: Esma Yıldız Özkan
^*Corresponding author: Esma Yıldız Özkan

Received: August 2022

Revised: October 2022

Early access: October 27, 2022

Published online: October 27, 2022

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Abstract

We investigate some new inequalities by estimating the rate of convergence by means of the complete modulus of continuity and a class of Lipschitz functions for the bivariate Bernstein-type operator including Bézier basis and present an example including numerical results comparing its rate of convergence. Moreover, we introduce its GBS (Generalized Boolean Sum) operator and obtain its rate of convergence with the help of the mixed modulus of continuity and Lipschitz class of Bögel continuous functions with exemplifying numerical results. Our research will demonstrate that the GBS operator possesses at least better numerical results than the bivariate Bernstein-type operator. All mentioned results point out the novelty of this study.

Keywords:

Mathematics Subject Classification: Primary: 41A25; Secondary: 41A36.

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Table 1. The approximation error by $ \mathcal{B}_{n,n}^{\lambda _{1},\lambda _{2}}\left(g\right) $ to $ g $ by means of the complete modulus of continuity for $ \lambda = 0.5 $

$ \lambda=0.5 $	$ n=40 $	$ n=30 $	$ n=20 $
$ \delta _{1}=\delta _{2} $	$ .5171701557\times 10^{-1} $	$ .5977184614\times 10^{-1} $	$ .7335943962\times 10^{-1} $
$ \omega \left( g;\delta _{1},\delta _{2}\right) $	$ .1007593815 $	$ .1159710188 $	$ .1413372719 $
$ E_{1}\left(\mathcal{B}_{n,n}^{\lambda _{1},\lambda _{2}}\left(g\right) ,g\right) $	$ .4030375260 $	$ .4638840752 $	$ .5653490876 $

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Table 2. The approximation error by $ \mathcal{B}_{n,n}^{\lambda _{1},\lambda _{2}}\left(g\right) $ to $ g $ by means of the complete modulus of continuity for $ \lambda = 0 $

$ \lambda=0 $	$ n=40 $	$ n=30 $	$ n=20 $
$ \delta _{1}=\delta _{2} $	$ .790569415\times 10^{-1} $	$ .9128709291\times 10^{-1} $	$ .11180339887 $
$ \omega \left( g;\delta _{1},\delta _{2}\right) $	$ .1518638830 $	$ .1742408526 $	$ .2111067978 $
$ E_{2}\left(\mathcal{B}_{n,n}^{\lambda _{1},\lambda _{2}}\left(g\right) ,g\right) $	$ .6074555320 $	$ .6969634104 $	$ .8444271912 $

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Table 3. The approximation error by $ \mathcal{B}_{n,n}^{\lambda _{1},\lambda _{2}}\left(g\right) $ to $ g $ by means of the Lipschitz functions on $ I_{R} $ for $ \lambda = 0.5 $ and $ \eta = 0.1 $

$ \lambda=0.5 $	$ n=40 $	$ n=30 $	$ n=20 $
$ \delta _{1}=\delta _{2} $	$ .5171701557\times 10^{-1} $	$ .5977184614\times 10^{-1} $	$ .7335943962\times 10^{-1} $
$ M_{g} $	$ .1822043754 $	$ .2037277262 $	$ .2383227509 $
$ E_{3}\left(\mathcal{B}_{n,n}^{\lambda _{1},\lambda _{2}}\left(g\right) ,g\right) $	$ .1007593815 $	$ .1159710187 $	$ .1413372719 $

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Table 4. The approximation error by $ \mathcal{B}_{n,n}^{\lambda _{1},\lambda _{2}}\left(g\right) $ to $ g $ by means of the Lipschitz functions on $ I_{R} $ for $ \lambda = 0 $ and $ \eta = 0.1 $

$ \lambda=0 $	$ n=40 $	$ n=30 $	$ n=20 $
$ \delta _{1}=\delta _{2} $	$ .790569415\times 10^{-1} $	$ .9128709291\times 10^{-1} $	$ 0.1118033988 $
$ M_{g} $	$ .2522705177 $	$ .2812341864 $	$ .3271984342 $
$ E_{4}\left(\mathcal{B}_{n,n}^{\lambda _{1},\lambda _{2}}\left(g\right) ,g\right) $	$ .1518638830 $	$ .1742408524 $	$ .2111067977 $

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Table 5. The approximation error by $ \mathcal{G}_{n,n}^{\lambda _{1},\lambda _{2}}\left(g\right) $ to $ g $ by means of the mixed modulus of continuity for $ \lambda = 0.5 $

$ \lambda=0.5 $	$ n=40 $	$ n=30 $	$ n=20 $
$ \delta _{1}=\delta _{2} $	$ .5171701557\times 10^{-1} $	$ .5977184614\times 10^{-1} $	$ .7335943962\times 10^{-1} $
$ \omega _{mixed}\left( g;\delta _{1},\delta _{2}\right) $	$ .2674649699\times 10^{-2} $	$ .3572673591\times 10^{-2} $	$ .5381607381\times 10^{-2} $
$ E_{5}\left(\mathcal{G}_{n,n}^{\lambda _{1},\lambda _{2}}\left(g\right) ,g\right) $	$ .1069859880\times 10^{-1} $	$ .1429069436\times 10^{-1} $	$ .2152642952\times 10^{-1} $

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Table 6. The approximation error by $ \mathcal{G}_{n,n}^{\lambda _{1},\lambda _{2}}\left(g\right) $ to $ g $ by means of the mixed modulus of continuity for $ \lambda = 0 $

$ \lambda=0 $	$ n=40 $	$ n=30 $	$ n=20 $
$ \delta _{1}=\delta _{2} $	$ .7905694150\times 10^{-1} $	$ .9128709291\times 10^{-1} $	$ .2738612787 $
$ \omega _{mixed}\left( g;\delta _{1},\delta _{2}\right) $	$ .6250000000\times 10^{-2} $	$ .8333333333\times 10^{-2} $	$ .1250000000\times 10^{-1} $
$ E_{6}\left(\mathcal{G}_{n,n}^{\lambda _{1},\lambda _{2}}\left(g\right) ,g\right) $	$ .2500000000\times 10^{-1} $	$ .3333333333\times 10^{-1} $	$ .5000000000\times 10^{-1} $

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Table 7. The approximation error by $ \mathcal{G}_{n,n}^{\lambda _{1},\lambda _{2}}\left(g\right) $ to $ g $ by means of the Lipschitz-type Bögel-continuous functions on $ I_{R} $ for $ \lambda = 0.5 $ and $ \gamma = 0.1 $

$ \lambda=0.5 $	$ n=40 $	$ n=30 $	$ n=20 $
$ \delta _{1}=\delta _{2} $	$ .5171701557\times 10^{-1} $	$ .5977184614\times 10^{-1} $	$ .7335943962\times 10^{-1} $
$ M_{g} $	$ .4836600531\times 10^{-2} $	$ .6276159987\times 10^{-2} $	$ .9074460391\times 10^{-2} $
$ E_{7}\left(\mathcal{G}_{n,n}^{\lambda _{1},\lambda _{2}}\left(g\right) ,g\right) $	$ .2674649699\times 10^{-2} $	$ .3572673591\times 10^{-2} $	$ .5381607382\times 10^{-2} $

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Table 8. The approximation error by $ \mathcal{G}_{n,n}^{\lambda _{1},\lambda _{2}}\left(g\right) $ to $ g $ by means of the Lipschitz-type Bögel-continuous functions on $ I_{R} $ for $ \lambda = 0 $ and $ \gamma = 0.1 $

$ \lambda=0 $	$ n=40 $	$ n=30 $	$ n=20 $
$ \delta _{1}=\delta _{2} $	$ .7905694150\times 10^{-1} $	$ .9128709291 \times 10^{-1} $	$ .1118033988 $
$ M_{g} $	$ .1038226275\times 10^{-1} $	$ .1345045199\times 10^{-1} $	$ .1937398734\times 10^{-1} $
$ E_{8}\left(\mathcal{G}_{n,n}^{\lambda _{1},\lambda _{2}}\left(g\right) ,g\right) $	$ .6250000001\times 10^{-2} $	$ .8333333331\times 10^{-2} $	$ .1250000000\times 10^{-1} $

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