doi: 10.3934/mfc.2022011
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On Szász-Durrmeyer type modification using Gould Hopper polynomials

1. 

Department of Applied Sciences and Humanities, Institute of Engineering and Technology, Dr. A.P.J. Abdul Kalam Technical University, Lucknow-226021(Uttar Pradesh), India

2. 

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667(Uttarakhand), India

*Corresponding author: Karunesh Kumar Singh

Received  November 2021 Revised  March 2022 Early access April 2022

In the present article, we study a generalization of Szász operators by Gould-Hopper polynomials. First, we obtain an estimate of error of the rate of convergence by these operators in terms of first order and second order moduli of continuity. Then, we derive a Voronovkaya-type theorem for these operators. Lastly, we derive Grüss-Voronovskaya type approximation theorem and Grüss-Voronovskaya type asymptotic result in quantitative form.

Citation: Karunesh Kumar Singh, Purshottam Narain Agrawal. On Szász-Durrmeyer type modification using Gould Hopper polynomials. Mathematical Foundations of Computing, doi: 10.3934/mfc.2022011
References:
[1]

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.

[2]

A. M. AcuH. Gonska and I. Raşa, Grüss-type and Ostrowski-type inequalities in approximation theory, Ukrainian Math. J., 63 (2011), 843-864.  doi: 10.1007/s11253-011-0548-2.

[3]

F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and Its Applications, De Gruyter Studies in Mathematics, 17, Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110884586.

[4]

D. Andrica and C. Badea, Grüss' inequality for positive linear functionals, Period. Math. Hungar., 19 (1988), 155-167.  doi: 10.1007/BF01848061.

[5]

P. Cardaliaguet and G. Euvrard, Approximation of a function and its derivative with a neural network, Neural Netw., 5 (1992), 207-220.  doi: 10.1016/S0893-6080(05)80020-6.

[6] F. Cucker and D.-X. Zhou, Learning Theory: An Approximation Theory Viewpoint, Cambridge Monographs on Applied and Computational Mathematics, 24, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511618796.
[7]

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

[8]

G. Grüss, Über das Maximum des absoluten Betrages von $(1/(b-a))\int_a^bf(x)g(x)dx-(1/(b-a)^2)\int_a^bf(x)dx \int_a^b g(x)dx$, Math. Z., 39 (1935), 215-226.  doi: 10.1007/BF01201355.

[9]

X. GuoL. Li and Q. Wu, Modeling interactive components by coordinate kernel polynomial models, Math. Found. Comput., 3 (2020), 263-277.  doi: 10.3934/mfc.2020010.

[10]

Z.-C. GuoD.-H. XiangX. Guo and D.-X. Zhou, Thresholded spectral algorithms for sparse approximations, Anal. Appl. (Singap.), 15 (2017), 433-455.  doi: 10.1142/S0219530517500026.

[11]

P. Gupta and P. N. Agrawal, Quantitative Voronovskaya and Grüss Voronovskaya-type theorems for operators of Kantorovich type involving multiple Appell polynomials, Iran. J. Sci. Technol. Trans. A Sci., 43 (2019), 1679-1687.  doi: 10.1007/s40995-018-0613-x.

[12]

V. Gupta, T. M. Rassias, P. N. Agrawal and A. M. Acu, Univariate Grüss-and Ostrowski-type inequalities for positive linear operators, in Recent Advances in Constructive Approximation Theory, Springer Optimization and Its Applications, 138, Springer, Cham, 2018,135–161. doi: 10.1007/978-3-319-92165-5_5.

[13]

V. Gupta and H. M. Srivastava, A general family of the Srivastava-Gupta operators preserving linear functions, Eur. J. Pure Appl. Math., 11 (2018), 575-579.  doi: 10.29020/nybg.ejpam.v11i3.3314.

[14]

N. İspir, On modified Baskakov opertors on weighted spaces, Turkish J. Math., 25 (2001), 355-365. 

[15]

A. Jakimovski and D. Leviatan, Generalized Szász operators for the approximation in the infinite interval, Mathematica (Cluj), 11 (1969), 97-103. 

[16]

A. KajlaS. Deshwal and P. N. Agrawal, Quantitative Voronovskaya and Grüss-Voronovskaya type theorems for Jain-Durrmeyer operators of blending type, Anal. Math. Phys., 9 (2019), 1241-1263.  doi: 10.1007/s13324-018-0229-5.

[17]

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

[18]

H. M. Srivastava and V. Gupta, A certain family of summation-integral type operators, Math. Comput. Modelling, 37 (2003), 1307-1315.  doi: 10.1016/S0895-7177(03)90042-2.

[19]

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

[20]

S. VarmaS. Sucu and G. İçöz, Generalization of Szász operators involving Brenke type polynomials, Comput. Math. Appl., 64 (2012), 121-127.  doi: 10.1016/j.camwa.2012.01.025.

[21]

S. Varma and F. Taşdelen, On a generalization of Szász-Durrmeyer operators with some orthogonal polynomials, Stud. Univ. Babeş-Bolyai Math., 58 (2013), 225–232.

[22]

D.-X. Zhou, Deep distributed convolutional neural networks: Universality, Anal. Appl. (Singap.), 16 (2018), 895-919.  doi: 10.1142/S0219530518500124.

show all references

References:
[1]

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.

[2]

A. M. AcuH. Gonska and I. Raşa, Grüss-type and Ostrowski-type inequalities in approximation theory, Ukrainian Math. J., 63 (2011), 843-864.  doi: 10.1007/s11253-011-0548-2.

[3]

F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and Its Applications, De Gruyter Studies in Mathematics, 17, Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110884586.

[4]

D. Andrica and C. Badea, Grüss' inequality for positive linear functionals, Period. Math. Hungar., 19 (1988), 155-167.  doi: 10.1007/BF01848061.

[5]

P. Cardaliaguet and G. Euvrard, Approximation of a function and its derivative with a neural network, Neural Netw., 5 (1992), 207-220.  doi: 10.1016/S0893-6080(05)80020-6.

[6] F. Cucker and D.-X. Zhou, Learning Theory: An Approximation Theory Viewpoint, Cambridge Monographs on Applied and Computational Mathematics, 24, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511618796.
[7]

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

[8]

G. Grüss, Über das Maximum des absoluten Betrages von $(1/(b-a))\int_a^bf(x)g(x)dx-(1/(b-a)^2)\int_a^bf(x)dx \int_a^b g(x)dx$, Math. Z., 39 (1935), 215-226.  doi: 10.1007/BF01201355.

[9]

X. GuoL. Li and Q. Wu, Modeling interactive components by coordinate kernel polynomial models, Math. Found. Comput., 3 (2020), 263-277.  doi: 10.3934/mfc.2020010.

[10]

Z.-C. GuoD.-H. XiangX. Guo and D.-X. Zhou, Thresholded spectral algorithms for sparse approximations, Anal. Appl. (Singap.), 15 (2017), 433-455.  doi: 10.1142/S0219530517500026.

[11]

P. Gupta and P. N. Agrawal, Quantitative Voronovskaya and Grüss Voronovskaya-type theorems for operators of Kantorovich type involving multiple Appell polynomials, Iran. J. Sci. Technol. Trans. A Sci., 43 (2019), 1679-1687.  doi: 10.1007/s40995-018-0613-x.

[12]

V. Gupta, T. M. Rassias, P. N. Agrawal and A. M. Acu, Univariate Grüss-and Ostrowski-type inequalities for positive linear operators, in Recent Advances in Constructive Approximation Theory, Springer Optimization and Its Applications, 138, Springer, Cham, 2018,135–161. doi: 10.1007/978-3-319-92165-5_5.

[13]

V. Gupta and H. M. Srivastava, A general family of the Srivastava-Gupta operators preserving linear functions, Eur. J. Pure Appl. Math., 11 (2018), 575-579.  doi: 10.29020/nybg.ejpam.v11i3.3314.

[14]

N. İspir, On modified Baskakov opertors on weighted spaces, Turkish J. Math., 25 (2001), 355-365. 

[15]

A. Jakimovski and D. Leviatan, Generalized Szász operators for the approximation in the infinite interval, Mathematica (Cluj), 11 (1969), 97-103. 

[16]

A. KajlaS. Deshwal and P. N. Agrawal, Quantitative Voronovskaya and Grüss-Voronovskaya type theorems for Jain-Durrmeyer operators of blending type, Anal. Math. Phys., 9 (2019), 1241-1263.  doi: 10.1007/s13324-018-0229-5.

[17]

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

[18]

H. M. Srivastava and V. Gupta, A certain family of summation-integral type operators, Math. Comput. Modelling, 37 (2003), 1307-1315.  doi: 10.1016/S0895-7177(03)90042-2.

[19]

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

[20]

S. VarmaS. Sucu and G. İçöz, Generalization of Szász operators involving Brenke type polynomials, Comput. Math. Appl., 64 (2012), 121-127.  doi: 10.1016/j.camwa.2012.01.025.

[21]

S. Varma and F. Taşdelen, On a generalization of Szász-Durrmeyer operators with some orthogonal polynomials, Stud. Univ. Babeş-Bolyai Math., 58 (2013), 225–232.

[22]

D.-X. Zhou, Deep distributed convolutional neural networks: Universality, Anal. Appl. (Singap.), 16 (2018), 895-919.  doi: 10.1142/S0219530518500124.

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