doi: 10.3934/mfc.2022015
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Direct and converse theorems for King type operators

Babeş-Bolyai University, Department of Mathematics, 1, M. Kogălniceanu st., 400084 Cluj-Napoca, Romania

*Corresponding author: Zoltán Finta

Received  April 2022 Revised  May 2022 Early access June 2022

For a sequence of King type operators which preserve the functions $ e_0(x)=1 $ and $ e_j(x)=x^j $, we establish a direct approximation theorem via the first order Ditzian-Totik modulus of smoothness, and a converse result of Berens-Lorentz type.

Citation: Zoltán Finta. Direct and converse theorems for King type operators. Mathematical Foundations of Computing, doi: 10.3934/mfc.2022015
References:
[1]

T. Acar, M. Cappelletti Montano, P. Garrancho and V. Leonessa, On sequences of J. P. King-type operators, J. Funct. Spaces, 2019 (2019), 12 pp. doi: 10.1155/2019/2329060.

[2]

A.-M. AcuH. Gonska and M. Heilmann, Remarks on a Bernstein-type operator of Aldaz, Kounchev and Render, J. Numer. Anal. Approx. Theory, 50 (2021), 3-11. 

[3]

J. M. AldazO. Kounchev and H. Render, Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces, Numer. Math., 114 (2009), 1-25.  doi: 10.1007/s00211-009-0248-0.

[4]

S. Bernstein, Démonstration du théorème de Weierstrass, fondeé sur le calcul des probabilités, Commun. Soc. Math. Kharkow, 13 (1912-13), 1-2. 

[5]

M. M. Birou, A proof of a conjecture about the asymptotic formula of a Bernstein type operator, Results Math., 72 (2017), 1129-1138.  doi: 10.1007/s00025-016-0608-x.

[6]

D. Cárdenas-Morales, P. Garrancho and I. Raşa, Asymptotic formulae via a Korovkin-type result, Abstr. Appl. Anal., 2012 (2012), 12 pp. doi: 10.1155/2012/217464.

[7]

R. A. DeVore and G. G. Lorentz, Constructive Approximation, Fundamental Principles of Mathematical Sciences, 303, Springer-Verlag, Berlin, 1993.

[8]

Z. Ditzian and V. Totik, Moduli of Smoothness, Springer Series in Computational Mathematics, 9, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4778-4.

[9]

M. Felten, Local and global approximation theorems for positive linear operators, J. Approx. Theory, 94 (1998), 396-419.  doi: 10.1006/jath.1998.3212.

[10]

Z. Finta, Bernstein type operators having 1 and xj as fixed points, Cent. Eur. J. Math., 11 (2013), 2257-2261.  doi: 10.2478/s11533-013-0310-0.

[11]

Z. Finta, Direct and converse theorems for King operators, Acta Univ. Sapientiae Math., 12 (2020), 85-96.  doi: 10.2478/ausm-2020-0005.

[12]

Z. Finta, New properties of King's operators, Positivity, 17 (2013), 101-109.  doi: 10.1007/s11117-011-0151-7.

[13]

Z. Finta, A quantitative variant of Voronovskaja's theorem for King-type operators, Constr. Math. Anal., 2 (2019), 124-129.  doi: 10.33205/cma.553427.

[14]

I. Gavrea and M. Ivan, Complete asymptotic expansions related to conjecture on a Voronovskaja-type theorem, J. Math. Anal. Appl., 458 (2018), 452-463.  doi: 10.1016/j.jmaa.2017.09.011.

[15]

J. P. King, Positive linear operators which preserve x2, Acta Math. Hungar, 99 (2003), 203-208.  doi: 10.1023/A:1024571126455.

show all references

References:
[1]

T. Acar, M. Cappelletti Montano, P. Garrancho and V. Leonessa, On sequences of J. P. King-type operators, J. Funct. Spaces, 2019 (2019), 12 pp. doi: 10.1155/2019/2329060.

[2]

A.-M. AcuH. Gonska and M. Heilmann, Remarks on a Bernstein-type operator of Aldaz, Kounchev and Render, J. Numer. Anal. Approx. Theory, 50 (2021), 3-11. 

[3]

J. M. AldazO. Kounchev and H. Render, Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces, Numer. Math., 114 (2009), 1-25.  doi: 10.1007/s00211-009-0248-0.

[4]

S. Bernstein, Démonstration du théorème de Weierstrass, fondeé sur le calcul des probabilités, Commun. Soc. Math. Kharkow, 13 (1912-13), 1-2. 

[5]

M. M. Birou, A proof of a conjecture about the asymptotic formula of a Bernstein type operator, Results Math., 72 (2017), 1129-1138.  doi: 10.1007/s00025-016-0608-x.

[6]

D. Cárdenas-Morales, P. Garrancho and I. Raşa, Asymptotic formulae via a Korovkin-type result, Abstr. Appl. Anal., 2012 (2012), 12 pp. doi: 10.1155/2012/217464.

[7]

R. A. DeVore and G. G. Lorentz, Constructive Approximation, Fundamental Principles of Mathematical Sciences, 303, Springer-Verlag, Berlin, 1993.

[8]

Z. Ditzian and V. Totik, Moduli of Smoothness, Springer Series in Computational Mathematics, 9, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4778-4.

[9]

M. Felten, Local and global approximation theorems for positive linear operators, J. Approx. Theory, 94 (1998), 396-419.  doi: 10.1006/jath.1998.3212.

[10]

Z. Finta, Bernstein type operators having 1 and xj as fixed points, Cent. Eur. J. Math., 11 (2013), 2257-2261.  doi: 10.2478/s11533-013-0310-0.

[11]

Z. Finta, Direct and converse theorems for King operators, Acta Univ. Sapientiae Math., 12 (2020), 85-96.  doi: 10.2478/ausm-2020-0005.

[12]

Z. Finta, New properties of King's operators, Positivity, 17 (2013), 101-109.  doi: 10.1007/s11117-011-0151-7.

[13]

Z. Finta, A quantitative variant of Voronovskaja's theorem for King-type operators, Constr. Math. Anal., 2 (2019), 124-129.  doi: 10.33205/cma.553427.

[14]

I. Gavrea and M. Ivan, Complete asymptotic expansions related to conjecture on a Voronovskaja-type theorem, J. Math. Anal. Appl., 458 (2018), 452-463.  doi: 10.1016/j.jmaa.2017.09.011.

[15]

J. P. King, Positive linear operators which preserve x2, Acta Math. Hungar, 99 (2003), 203-208.  doi: 10.1023/A:1024571126455.

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