doi: 10.3934/mfc.2022002
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Rate of convergence of Stancu type modified $ q $-Gamma operators for functions with derivatives of bounded variation

1. 

Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics, 14280 Bolu, Turkey

2. 

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India

* Corresponding author: Purshottam Narain Agrawal

Received  September 2021 Revised  January 2022 Early access February 2022

Recently, Karsli [15] estimated the convergence rate of the $ q $-Bernstein-Durrmeyer operators for functions whose $ q $-derivatives are of bounded variation on the interval $ [0, 1] $. Inspired by this study, in the present paper we deal with the convergence rate of a $ q $- analogue of the Stancu type modified Gamma operators, defined by Karsli et al. [17], for the functions $ \varphi $ whose $ q $-derivatives are of bounded variation on the interval $ [0, \infty ). $ We present the approximation degree for the operator $ \left( { \mathfrak{S}}_{n, \ell, q}^{(\alpha , \beta )} { \varphi}\right)(\mathfrak{z}) $ at those points $ \mathfrak{z} $ at which the one sided q-derivatives$ {D}_{q}^{+}{ \varphi(\mathfrak{z})\; and\; D} _{q}^{-}{ \varphi(\mathfrak{z})} $ exist.

Citation: Harun Karsli, Purshottam Narain Agrawal. Rate of convergence of Stancu type modified $ q $-Gamma operators for functions with derivatives of bounded variation. Mathematical Foundations of Computing, doi: 10.3934/mfc.2022002
References:
[1]

A.-M. AcuC. V. MuraruD. F. Sofonea and V. A. Radu, Some approximation properties of a Durrmeyer variant of $q$-Bernstein–Schurer operators, Math. Methods Appl. Sci., 39 (2016), 5636-5650.  doi: 10.1002/mma.3949.

[2]

T. Acar and A. Aral, On pointwise convergence of $q$-Bernstein operators and their $q$-derivatives, Numer. Funct. Anal. Optim., 36 (2015), 287-304.  doi: 10.1080/01630563.2014.970646.

[3]

P. N. Agrawal, H. Karsli and M. Goyal, Szász-Baskakov type operators based on $q$-integers, J. Inequal. Appl., 2014 (2014), 441, 18 pp. doi: 10.1186/1029-242X-2014-441.

[4]

A. Aral, V. Gupta and R. P. Agarwal, Applications of $q$- Calculus in Operator Theory, Vol. XII. Springer: New York, 2013. doi: 10.1007/978-1-4614-6946-9.

[5]

Q.-B. Cai and X.-M. Zeng, On the convergence of a kind of $q$-Gamma operators, J. Inequal. Appl., 2013 (2013), 105, 9 pp. doi: 10.1186/1029-242X-2013-105.

[6]

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.

[7]

W. Chen and S. Guo, On the rate of convergence of the Gamma operator for functions of bounded variation, Approx. Theory Appl., 1 (1985), 85-96. 

[8] F. Cucker and D.-X. Zhou, Learning Theory, An Approximation Theory View Point, Cambridge University Press, 2007.  doi: 10.1017/CBO9780511618796.
[9]

R. J. Finkelstein, $q$-Uncertainty relations, Internat. J. Modern Phys. A., 13 (1998), 1795-1803.  doi: 10.1142/S0217751X98000780.

[10]

X. GuoL. Li and Q. Wu, Modeling interactive components by coordinate kernel polynomial models, Math. Found. Comp., 3 (2020), 263-277. 

[11]

V. Gupta and H. Karsli, Some approximation properties by $q$-Sz$\acute{a}$sz-Mirakyan-Baskakov-Stancu operators, Lobachevskii J. Math., 33 (2012), 175-182.  doi: 10.1134/S1995080212020138.

[12]

C.-L. Ho, On the use of Mellin transform to a class of $q$-difference-differential equations, Phys. Lett. A., 268 (2000), 217-223.  doi: 10.1016/S0375-9601(00)00191-2.

[13]

A. Izgi and I. Büyükyazici, Sınırsız Aralıklarda Yaklaș ım ve Yaklașım Hızı, Kastamonu Eğitim Dergisi Ekim., 11 (2003), 451–460.

[14]

V. Kac and P. Cheung, Quantum Calculus. Universitext, New York: Springer-Verlag, 2002. doi: 10.1007/978-1-4613-0071-7.

[15]

H. Karsli, Some approximation properties of $q$-Bernstein-Durrmeyer operators, Tbilisi Math. J., 12 (2019), 189-204.  doi: 10.32513/tbilisi/1578020576.

[16]

H. Karsli, On approximation to discrete q-derivatives of functions via $q$-Bernstein-Schurer operators, Math. Found. Comp., 4 (2021), 15-30.  doi: 10.3934/mfc.2020023.

[17]

H. KarsliP. N. Agrawal and M. Goyal, General Gamma type operators based on $q$-integers, Appl. Math. Comput., 251 (2015), 564-575.  doi: 10.1016/j.amc.2014.11.085.

[18]

H. Karsli and V. Gupta, Some approximation properties of $q$-Chlodowsky operators, Appl. Math. Comput., 195 (2008), 220-229.  doi: 10.1016/j.amc.2007.04.085.

[19]

H. Karsli and M. A. Özarslan, Direct local and global approximation results for operators of Gamma type, Hacet. J. Math. Stat., 39 (2010), 241-253. 

[20]

D. LeviJ. Negro and M. A. del Olmo, Discrete $q$-derivatives and symmetries of $q$-difference equations, J. Phys. A, 37 (2004), 3459-3473.  doi: 10.1088/0305-4470/37/10/010.

[21]

A. Lupaș, A $q$-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on Numerical and Statistical Calculus, 9 (1987), 85-92. 

[22]

A. Lupaș and M. Müller, Approximationseigenschaften der Gammaoperatoren, Math. Zeitschr., 98 (1967), 208-226.  doi: 10.1007/BF01112415.

[23]

L. C. Mao, Rate of convergence of Gamma type operator, Journal of Shangqiu Teachers College, 23 (2007), 49-52. 

[24]

S. M. Mazhar, Approximation by positive operators on infinite intervals, Math. Balkanica, 5 (1991), 99-104. 

[25]

K. Mezlini and N. Bettaibi, Generalized discrete $q$-Hermite I polynomials and $q$-deformed oscillator, Acta Math. Sci. Ser. B (Eng. Ed.), 38 (2018), 1411-1426.  doi: 10.1016/S0252-9602(18)30822-1.

[26]

G. M. Phillips, Bernstein polynomials based on $q$-integers, in The heritage of P. L. Chebyshev: a Festschrift in honour of the 70th birthday of T. J. Rivlin, Ann. Numer. Math., 4 (1997), 511-518. 

[27]

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

[28]

X.-M. Zeng, Approximation properties of Gamma operators, J. Math. Anal. Appl., 311 (2005), 389-401.  doi: 10.1016/j.jmaa.2005.02.051.

[29]

C. Zhao, W.-T. Cheng and X.-M. Zeng, Some approximation properties of a kind of $q$-Gamma-Stancu operators, J. Inequal. Appl., 2014 (2014), 94, 13 pp. doi: 10.1186/1029-242X-2014-94.

[30]

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

show all references

References:
[1]

A.-M. AcuC. V. MuraruD. F. Sofonea and V. A. Radu, Some approximation properties of a Durrmeyer variant of $q$-Bernstein–Schurer operators, Math. Methods Appl. Sci., 39 (2016), 5636-5650.  doi: 10.1002/mma.3949.

[2]

T. Acar and A. Aral, On pointwise convergence of $q$-Bernstein operators and their $q$-derivatives, Numer. Funct. Anal. Optim., 36 (2015), 287-304.  doi: 10.1080/01630563.2014.970646.

[3]

P. N. Agrawal, H. Karsli and M. Goyal, Szász-Baskakov type operators based on $q$-integers, J. Inequal. Appl., 2014 (2014), 441, 18 pp. doi: 10.1186/1029-242X-2014-441.

[4]

A. Aral, V. Gupta and R. P. Agarwal, Applications of $q$- Calculus in Operator Theory, Vol. XII. Springer: New York, 2013. doi: 10.1007/978-1-4614-6946-9.

[5]

Q.-B. Cai and X.-M. Zeng, On the convergence of a kind of $q$-Gamma operators, J. Inequal. Appl., 2013 (2013), 105, 9 pp. doi: 10.1186/1029-242X-2013-105.

[6]

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.

[7]

W. Chen and S. Guo, On the rate of convergence of the Gamma operator for functions of bounded variation, Approx. Theory Appl., 1 (1985), 85-96. 

[8] F. Cucker and D.-X. Zhou, Learning Theory, An Approximation Theory View Point, Cambridge University Press, 2007.  doi: 10.1017/CBO9780511618796.
[9]

R. J. Finkelstein, $q$-Uncertainty relations, Internat. J. Modern Phys. A., 13 (1998), 1795-1803.  doi: 10.1142/S0217751X98000780.

[10]

X. GuoL. Li and Q. Wu, Modeling interactive components by coordinate kernel polynomial models, Math. Found. Comp., 3 (2020), 263-277. 

[11]

V. Gupta and H. Karsli, Some approximation properties by $q$-Sz$\acute{a}$sz-Mirakyan-Baskakov-Stancu operators, Lobachevskii J. Math., 33 (2012), 175-182.  doi: 10.1134/S1995080212020138.

[12]

C.-L. Ho, On the use of Mellin transform to a class of $q$-difference-differential equations, Phys. Lett. A., 268 (2000), 217-223.  doi: 10.1016/S0375-9601(00)00191-2.

[13]

A. Izgi and I. Büyükyazici, Sınırsız Aralıklarda Yaklaș ım ve Yaklașım Hızı, Kastamonu Eğitim Dergisi Ekim., 11 (2003), 451–460.

[14]

V. Kac and P. Cheung, Quantum Calculus. Universitext, New York: Springer-Verlag, 2002. doi: 10.1007/978-1-4613-0071-7.

[15]

H. Karsli, Some approximation properties of $q$-Bernstein-Durrmeyer operators, Tbilisi Math. J., 12 (2019), 189-204.  doi: 10.32513/tbilisi/1578020576.

[16]

H. Karsli, On approximation to discrete q-derivatives of functions via $q$-Bernstein-Schurer operators, Math. Found. Comp., 4 (2021), 15-30.  doi: 10.3934/mfc.2020023.

[17]

H. KarsliP. N. Agrawal and M. Goyal, General Gamma type operators based on $q$-integers, Appl. Math. Comput., 251 (2015), 564-575.  doi: 10.1016/j.amc.2014.11.085.

[18]

H. Karsli and V. Gupta, Some approximation properties of $q$-Chlodowsky operators, Appl. Math. Comput., 195 (2008), 220-229.  doi: 10.1016/j.amc.2007.04.085.

[19]

H. Karsli and M. A. Özarslan, Direct local and global approximation results for operators of Gamma type, Hacet. J. Math. Stat., 39 (2010), 241-253. 

[20]

D. LeviJ. Negro and M. A. del Olmo, Discrete $q$-derivatives and symmetries of $q$-difference equations, J. Phys. A, 37 (2004), 3459-3473.  doi: 10.1088/0305-4470/37/10/010.

[21]

A. Lupaș, A $q$-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on Numerical and Statistical Calculus, 9 (1987), 85-92. 

[22]

A. Lupaș and M. Müller, Approximationseigenschaften der Gammaoperatoren, Math. Zeitschr., 98 (1967), 208-226.  doi: 10.1007/BF01112415.

[23]

L. C. Mao, Rate of convergence of Gamma type operator, Journal of Shangqiu Teachers College, 23 (2007), 49-52. 

[24]

S. M. Mazhar, Approximation by positive operators on infinite intervals, Math. Balkanica, 5 (1991), 99-104. 

[25]

K. Mezlini and N. Bettaibi, Generalized discrete $q$-Hermite I polynomials and $q$-deformed oscillator, Acta Math. Sci. Ser. B (Eng. Ed.), 38 (2018), 1411-1426.  doi: 10.1016/S0252-9602(18)30822-1.

[26]

G. M. Phillips, Bernstein polynomials based on $q$-integers, in The heritage of P. L. Chebyshev: a Festschrift in honour of the 70th birthday of T. J. Rivlin, Ann. Numer. Math., 4 (1997), 511-518. 

[27]

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

[28]

X.-M. Zeng, Approximation properties of Gamma operators, J. Math. Anal. Appl., 311 (2005), 389-401.  doi: 10.1016/j.jmaa.2005.02.051.

[29]

C. Zhao, W.-T. Cheng and X.-M. Zeng, Some approximation properties of a kind of $q$-Gamma-Stancu operators, J. Inequal. Appl., 2014 (2014), 94, 13 pp. doi: 10.1186/1029-242X-2014-94.

[30]

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

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