Convergence of \(\lambda\)-Bernstein - Kantorovich operators in the \(L_p\)- norm

Purshottam N. Agrawal; Behar Baxhaku

doi:10.33993/jnaat531-1374

Authors

Purshottam N. Agrawal Indian Institute of Technology, Roorkee, India https://orcid.org/0000-0003-3029-6896
Behar Baxhaku University of Prishtina "Hasan Prishtina", Kosovo https://orcid.org/0000-0002-8990-1440

DOI:

https://doi.org/10.33993/jnaat531-1374

Abstract views: 155

Abstract

We show the convergence of \(\lambda\)-Bernstein - Kantorovich operators defined by Acu et al. [J. Ineq. Appl. 2018], for functions in \(L_p[0,1],\, p\geq 1\). We also determine the convergence rate via integral modulus of smoothness.

Downloads

Download data is not yet available.

References

A. M. Acu, N. Manav and D. F. Sofonea, Approximation properties of λ-Kantorovich operators, J. Inequal. Appl. 202 (2018). DOI: https://doi.org/10.1186/s13660-018-1795-7

P. N. Agrawal, B. Baxhaku and R. Shukla, A new kind of bi-variate λ-Bernstein-Kantorovich type operator with shifted knots and its associated GBS form, Math. Found. Comput. 5, no.3 (2022), pp.157-172. DOI: https://doi.org/10.3934/mfc.2021025

R. Aslan, Approximation properties of univariate and bivariate new class λ-Bernstein–Kantorovich operators and its associated GBS operators, Comput. Appl. Math. 42, art. no. 34 (2023). https://doi.org/10.1007/s40314-022-02182-w DOI: https://doi.org/10.1007/s40314-022-02182-w

S. N. Bernstein, Demonstration du theoreme de Weierstrass fondee sur le calcul des probabilites, Commun. Soc. Math. Kharkow 2, no. 13 (1912), pp. 1-2.

M. Bodur, N. Manav and F. Tas ̧delen, Approximation properties of λ-Bernstein-Kantorovich-Stancu operators, Math. Slovaca 72, no.1 (2022), pp 141-152. https://doi.org/10.1515/ms-2022-0010 DOI: https://doi.org/10.1515/ms-2022-0010

Q. B. Cai, B. Y. Lian and G. Zhou, Approximation properties of λ-Bernstein operators, J. Inequal. Appl. 61 (2018). DOI: https://doi.org/10.1186/s13660-018-1653-7

R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin Heidelberg New York, 1993. DOI: https://doi.org/10.1007/978-3-662-02888-9

Z. Ditzian and K. Ivanov, Bernstein-type operators and their derivatives, J. Approx. Theory 56, no.1 (1989), pp. 72-90. https://doi.org/10.1016/0021-9045(89)90134-2 DOI: https://doi.org/10.1016/0021-9045(89)90134-2

S. Guo, C. Li, X. Liu and Z. Song, Pointwise approximation for linear combinations of Bernstein operators, J. Approx. Theory 107, no. 1 (2000), pp. 109-120. DOI: https://doi.org/10.1006/jath.2000.3504

A. Kumar, Approximation properties of generalized λ-Bernstein-Kantorovich type operators, Rend. Circ. Mat. Palermo, II. Ser, 70, 2021, pp.505-520. https://doi.org/10.1007/s12215-020-00509-2 DOI: https://doi.org/10.1007/s12215-020-00509-2

M. Mursaleen, K. J. Ansari and A. Khan, On (p, q)-analogue of Bernstein operators, Appl. Math. Comput. 266 (2015), pp.874-882. DOI: https://doi.org/10.1016/j.amc.2015.04.090

G. Nowak, Approximation properties for generalized q-Bernstein polynomials, J. Math. Anal. Appl. 350 (2009), pp.50–55. https://doi.org/10.1016/j.jmaa.2008.09.003 DOI: https://doi.org/10.1016/j.jmaa.2008.09.003

M. A. Ozarslan and O. Duman, Smoothness properties of modified Bernstein - Kantorovich operators, Numer. Funct. Anal. Optim. 37, no.1 (2016), pp.92-105. https://doi.org/10.1080/01630563.2015.1079219 DOI: https://doi.org/10.1080/01630563.2015.1079219

S. Rahman, M. Mursaleen and A. M. Acu, Approximation properties of λ-Bernstein-Kantorovich operators with shifted knots, Math. Methods. Appl. Sci. 42 (2019), pp.4042-4053. https://doi.org/10.1002/mma.5632 DOI: https://doi.org/10.1002/mma.5632

S. Sucu and E. Ibikli, Approximation by means of Kantorovich-Stancu type operators, Numer. Funct. Anal. Optim. 34, no.5 (2013), pp. 557-575. https://doi.org/10.1080/01630563.2012.716806 DOI: https://doi.org/10.1080/01630563.2012.716806

Z. Ye, X. Long and X. M. Zeng, Adjustment algorithms for Bezier curve and surface, In: International Conference on Computer Science and Education (2010), pp. 1712-1716.

X. M. Zeng and F.F. Cheng, On the rates of approximation of Bernstein type operators, J. Approx. Theory 109, no.2 (2001), pp.242-256. https://doi.org/10.1006/jath.2000.3538 DOI: https://doi.org/10.1006/jath.2000.3538

A. Zygmund, Trigonometric Series I, II, Cambridge University Press, Cambridge, U.K., 1959.