doi: 10.3934/mfc.2021027
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Convergence of modified Szász-Mirakyan-Durrmeyer operators depending on certain parameters

1. 

Department of Mathematical Sciences, Baba Ghulam Shah Badshah University, Rajouri-185234, Jammu and Kashmir, India

2. 

Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

3. 

Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan

* Corresponding author: Mohammad Mursaleen

Received  July 2021 Revised  August 2021 Early access October 2021

Motivated by certain generalizations, in this paper we consider a new analogue of modified Szá sz-Mirakyan-Durrmeyer operators whose construction depends on a continuously differentiable, increasing and unbounded function $ \tau $ with extra parameters $ \mu $ and $ \lambda $. Depending on the selection of $ \mu $ and $ \lambda $, these operators are more flexible than the modified Szá sz-Mirakyan-Durrmeyer operators while retaining their approximation properties. For these operators we give weighted approximation, Voronovskaya type theorem and quantitative estimates for the local approximation.

Citation: Mohd Qasim, Mohd Shanawaz Mansoori, Asif Khan, Zaheer Abbas, Mohammad Mursaleen. Convergence of modified Szász-Mirakyan-Durrmeyer operators depending on certain parameters. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021027
References:
[1]

T. Acar, Asymptotic formulas for generalized Szász-Mirakyan operators, Applied Mathematics and Computation, 263 (2015), 223-239.  doi: 10.1016/j.amc.2015.04.060.  Google Scholar

[2]

T. AcarA. Aral and I. Raşa, The new forms of Voronovskaya's theorem in weighted spaces, Positivity, 20 (2016), 25-40.  doi: 10.1007/s11117-015-0338-4.  Google Scholar

[3]

T. AcarA. Aral and I. Raşa, Modified Bernstein-Durrmeyer operators, Gen. Math., 22 (2014), 27-41.   Google Scholar

[4]

T. AcarA. Aral and I. Raşa, Positive linear operators preserving $\tau $ and $\tau ^{2}$, Constructive Mathematical Analysis, 2 (2019), 98-102.  doi: 10.33205/cma.547221.  Google Scholar

[5]

T. AcarS. A. Mohiudine and M. Mursaleen, Approximation by $ (p, q)$-Baskakov Durrmeyer Stancu operators, Complex Anal. Oper. Theory., 12 (2018), 1453-1468.  doi: 10.1007/s11785-016-0633-5.  Google Scholar

[6]

T. Acar and G. Ulusoy, Approximation by modified Szá sz-Durrmeyer operators, Period Math. Hung., 72 (2016), 64-75.  doi: 10.1007/s10998-015-0091-2.  Google Scholar

[7]

O. Agratini, On the rate of convergence of a positive approximation process, Nihonkai Math. J., 11 (2000), 47-56.   Google Scholar

[8]

A. Aral and V. Gupta, Generalized Szász Durrmeyer operators, Lobachevskii J. Math., 32 (2011), 23-31.  doi: 10.1134/S1995080211010033.  Google Scholar

[9]

A. AralD. Inoan and I. Raşa, On the generalized Szá sz-Mirakyan operators, Results Math., 65 (2014), 441-452.  doi: 10.1007/s00025-013-0356-0.  Google Scholar

[10]

S. N. Bernstein, Démonstation du théorème de Weierstrass fondée sur le calcul de probabilités, Commun. Soc. Math. Kharkow, 13 (1912-1913), 1-2.   Google Scholar

[11]

D. Cárdenas-MoralesP. Garrancho and I. Raşa, Bernstein-type operators which preserve polynomials, Comput. Math. Appl., 62 (2011), 158-163.  doi: 10.1016/j.camwa.2011.04.063.  Google Scholar

[12]

R. A. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren Math. Wiss. [Fundamental principales of Mathematical Sciences], Springer-Verlag, Berlin, 1993.  Google Scholar

[13]

A. D. Gadzhiev, A problem on the convergence of a sequence of positive linear operators on unbounded sets, and theorems that are analogous to P. P. Korovkin's theorem, Dokl. Akad. Nauk SSSR, 218 (1974), (Russian), 1001-1004.  Google Scholar

[14]

A. D. Gadzhiev and A. Aral, The estimates of approximation by using a new type of weighted modulus of continuity, Comp. Math. Appl., 54 (2007), 127-135.  doi: 10.1016/j.camwa.2007.01.017.  Google Scholar

[15]

A. Holhoş, Quantitative estimates for positive linear operators in weighted spaces, General Math., 16 (2008), 99-110.   Google Scholar

[16]

A. KilicmanM. Ayman Mursaleen and A. A. H. A. Al-Abied, Stancu type Baskakov-Durrmeyer operators and approximation properties, Mathematics, 8 (2020), 1164.  doi: 10.3390/math8071164.  Google Scholar

[17]

S. M. Mazhar and V. Totik, Approximation by modified Szász operators, Acta Sci. Math. (Szeged), 49 (1985), 257-269.   Google Scholar

[18]

V. N. Mishra and S. Pandey, On Chlodowsky variant of $(p, q) $ Kantorovich-Stancu-Schurer operators, Int. J. of Anal. Appl., 11 (2016), 28-39.   Google Scholar

[19]

S. A. Mohiuddine, T. Acar and M. A. Alghamdi, Genuine modified Bernstein-Durrmeyer operators, J. Ineq. Appl., 2018 (2018), Paper No. 104, 13 pp. doi: 10.1186/s13660-018-1693-z.  Google Scholar

[20]

M. Mursaleen and M. Ahasan, The Dunkl generalization of Stancu type $q$-Szász-Mirakyan- Kantrovich operators and some approximation results, Carpathian J. Math., 34 (2018), 363-370.  doi: 10.37193/CJM.2018.03.11.  Google Scholar

[21]

M. Mursaleen and T. Khan, On approximation by Stancu type Jakimovski-Leviatan-Durrmeyer operators, Azerbaijan Jour. Math., 7 (2017), 16-26.   Google Scholar

[22]

N. Rao and A. Wafi, Bivariate-Schurer-Stancu operators based on $(p, q)$-integers, Filomat, 32 (2018), 1251-1258.  doi: 10.2298/FIL1804251R.  Google Scholar

[23]

D. D. Stancu, Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pure Appl., 13 (1968), 1173-1194.   Google Scholar

[24]

O. Szász, Generalization of S. Bernstein's polynomials to the infinite interval, J. Res. Nat. Bur. Standards, 45 (1950), 239-245.  doi: 10.6028/jres.045.024.  Google Scholar

[25]

K. Weierstrass, Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen, Sitzungsberichte der Akademiezu Berlin, (1885), 633–639,789–805 Google Scholar

show all references

References:
[1]

T. Acar, Asymptotic formulas for generalized Szász-Mirakyan operators, Applied Mathematics and Computation, 263 (2015), 223-239.  doi: 10.1016/j.amc.2015.04.060.  Google Scholar

[2]

T. AcarA. Aral and I. Raşa, The new forms of Voronovskaya's theorem in weighted spaces, Positivity, 20 (2016), 25-40.  doi: 10.1007/s11117-015-0338-4.  Google Scholar

[3]

T. AcarA. Aral and I. Raşa, Modified Bernstein-Durrmeyer operators, Gen. Math., 22 (2014), 27-41.   Google Scholar

[4]

T. AcarA. Aral and I. Raşa, Positive linear operators preserving $\tau $ and $\tau ^{2}$, Constructive Mathematical Analysis, 2 (2019), 98-102.  doi: 10.33205/cma.547221.  Google Scholar

[5]

T. AcarS. A. Mohiudine and M. Mursaleen, Approximation by $ (p, q)$-Baskakov Durrmeyer Stancu operators, Complex Anal. Oper. Theory., 12 (2018), 1453-1468.  doi: 10.1007/s11785-016-0633-5.  Google Scholar

[6]

T. Acar and G. Ulusoy, Approximation by modified Szá sz-Durrmeyer operators, Period Math. Hung., 72 (2016), 64-75.  doi: 10.1007/s10998-015-0091-2.  Google Scholar

[7]

O. Agratini, On the rate of convergence of a positive approximation process, Nihonkai Math. J., 11 (2000), 47-56.   Google Scholar

[8]

A. Aral and V. Gupta, Generalized Szász Durrmeyer operators, Lobachevskii J. Math., 32 (2011), 23-31.  doi: 10.1134/S1995080211010033.  Google Scholar

[9]

A. AralD. Inoan and I. Raşa, On the generalized Szá sz-Mirakyan operators, Results Math., 65 (2014), 441-452.  doi: 10.1007/s00025-013-0356-0.  Google Scholar

[10]

S. N. Bernstein, Démonstation du théorème de Weierstrass fondée sur le calcul de probabilités, Commun. Soc. Math. Kharkow, 13 (1912-1913), 1-2.   Google Scholar

[11]

D. Cárdenas-MoralesP. Garrancho and I. Raşa, Bernstein-type operators which preserve polynomials, Comput. Math. Appl., 62 (2011), 158-163.  doi: 10.1016/j.camwa.2011.04.063.  Google Scholar

[12]

R. A. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren Math. Wiss. [Fundamental principales of Mathematical Sciences], Springer-Verlag, Berlin, 1993.  Google Scholar

[13]

A. D. Gadzhiev, A problem on the convergence of a sequence of positive linear operators on unbounded sets, and theorems that are analogous to P. P. Korovkin's theorem, Dokl. Akad. Nauk SSSR, 218 (1974), (Russian), 1001-1004.  Google Scholar

[14]

A. D. Gadzhiev and A. Aral, The estimates of approximation by using a new type of weighted modulus of continuity, Comp. Math. Appl., 54 (2007), 127-135.  doi: 10.1016/j.camwa.2007.01.017.  Google Scholar

[15]

A. Holhoş, Quantitative estimates for positive linear operators in weighted spaces, General Math., 16 (2008), 99-110.   Google Scholar

[16]

A. KilicmanM. Ayman Mursaleen and A. A. H. A. Al-Abied, Stancu type Baskakov-Durrmeyer operators and approximation properties, Mathematics, 8 (2020), 1164.  doi: 10.3390/math8071164.  Google Scholar

[17]

S. M. Mazhar and V. Totik, Approximation by modified Szász operators, Acta Sci. Math. (Szeged), 49 (1985), 257-269.   Google Scholar

[18]

V. N. Mishra and S. Pandey, On Chlodowsky variant of $(p, q) $ Kantorovich-Stancu-Schurer operators, Int. J. of Anal. Appl., 11 (2016), 28-39.   Google Scholar

[19]

S. A. Mohiuddine, T. Acar and M. A. Alghamdi, Genuine modified Bernstein-Durrmeyer operators, J. Ineq. Appl., 2018 (2018), Paper No. 104, 13 pp. doi: 10.1186/s13660-018-1693-z.  Google Scholar

[20]

M. Mursaleen and M. Ahasan, The Dunkl generalization of Stancu type $q$-Szász-Mirakyan- Kantrovich operators and some approximation results, Carpathian J. Math., 34 (2018), 363-370.  doi: 10.37193/CJM.2018.03.11.  Google Scholar

[21]

M. Mursaleen and T. Khan, On approximation by Stancu type Jakimovski-Leviatan-Durrmeyer operators, Azerbaijan Jour. Math., 7 (2017), 16-26.   Google Scholar

[22]

N. Rao and A. Wafi, Bivariate-Schurer-Stancu operators based on $(p, q)$-integers, Filomat, 32 (2018), 1251-1258.  doi: 10.2298/FIL1804251R.  Google Scholar

[23]

D. D. Stancu, Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pure Appl., 13 (1968), 1173-1194.   Google Scholar

[24]

O. Szász, Generalization of S. Bernstein's polynomials to the infinite interval, J. Res. Nat. Bur. Standards, 45 (1950), 239-245.  doi: 10.6028/jres.045.024.  Google Scholar

[25]

K. Weierstrass, Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen, Sitzungsberichte der Akademiezu Berlin, (1885), 633–639,789–805 Google Scholar

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