February  2021, 4(1): 15-30. doi: 10.3934/mfc.2020023

On approximation to discrete q-derivatives of functions via q-Bernstein-Schurer operators

Bolu Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics, 14030, Golkoy-Bolu, Turkey

* Corresponding author: Harun Karsli

Received  April 2020 Revised  September 2020 Published  November 2020

In the present paper, we shall investigate the pointwise approximation properties of the $ q- $analogue of the Bernstein-Schurer operators and estimate the rate of pointwise convergence of these operators to the functions $ f $ whose $ q- $derivatives are bounded variation on the interval $ [0,1+p]. $ We give an estimate for the rate of convergence of the operator $ \left( B_{n,p,q}f\right) $ at those points $ x $ at which the one sided $ q- $derivatives $D_{q}^{+}f(x) $ and $ D_{q}^{-}f(x) $ exist. We shall also prove that the operators $ \left( B_{n,p,q}f\right) (x) $ converge to the limit $ f(x) $. As a continuation of the very recent and initial study of the author deals with the pointwise approximation of the $ q- $Bernstein Durrmeyer operators [12] at those points $ x $ at which the one sided $ q- $derivatives $ D_{q}^{+}f(x) $ and $ D_{q}^{-}f(x) $ exist, this study provides (or presents) a forward work on the approximation of $ q $-analogue of the Schurer type operators in the space of $ D_{q}BV $.

Citation: Harun Karsli. On approximation to discrete q-derivatives of functions via q-Bernstein-Schurer operators. Mathematical Foundations of Computing, 2021, 4 (1) : 15-30. doi: 10.3934/mfc.2020023
References:
[1]

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.  Google Scholar

[2]

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.  Google Scholar

[3]

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

[4]

R. Bojanić and F. Chêng, Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation, J. Math. Anal. Appl., 141 (1989), 136-151.  doi: 10.1016/0022-247X(89)90211-4.  Google Scholar

[5]

R. Bojanić and F. Cheng, Rate of convergence of Hermite-Fejér polynomials for functions with derivatives of bounded variation, Acta Math. Hungar., 59 (1992), 91-102.  doi: 10.1007/BF00052094.  Google Scholar

[6]

R. Bojanić and M. Vuilleumier, On the rate of convergence of Fourier-Legendre series of functions of bounded variation, J. Approx. Theory, 31 (1981), 67-79.  doi: 10.1016/0021-9045(81)90031-9.  Google Scholar

[7]

F. H. Chêng, On the rate of convergence of Bernstein polynomials of functions of bounded variation, J. Approx. Theory, 39 (1983), 259-274.  doi: 10.1016/0021-9045(83)90098-9.  Google Scholar

[8]

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

[9]

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.  Google Scholar

[10]

F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-203.   Google Scholar

[11]

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

[12]

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

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

A. Lupaş, A q-analogue of the Bernstein operator, in Seminar on Numerical and Statistical Calculus, Univ. "Babeş-Bolyai", Cluj-Napoca, 1987, 85–92.  Google Scholar

[16]

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

[17]

C.-V. Muraru, Note on q-Bernstein-Schurer operators, Stud. Univ. Babeş-Bolyai Math., 56 (2011), 489–495.  Google Scholar

[18]

G. M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math., 4 (1997), 511-518.   Google Scholar

[19]

G. M. Phillips, On generalized Bernstein polynomials, in Numerical Analysis, World Sci. Publ., River Edge, NJ, 1996,263–269. doi: 10.1142/9789812812872_0018.  Google Scholar

[20]

M.-Y. Ren and X.-M. Zeng, On statistical approximation properties of modified q-Bernstein-Schurer operators, Bull. Korean Math. Soc., 50 (2013), 1145-1156.  doi: 10.4134/BKMS.2013.50.4.1145.  Google Scholar

[21]

J. Thomae, Beiträge zur Theorie der durch die Heinische Reihe: Darstellbaren Functionen, J. Reine Angew. Math., 70 (1869), 258-281.  doi: 10.1515/crll.1869.70.258.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

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.  Google Scholar

[3]

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

[4]

R. Bojanić and F. Chêng, Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation, J. Math. Anal. Appl., 141 (1989), 136-151.  doi: 10.1016/0022-247X(89)90211-4.  Google Scholar

[5]

R. Bojanić and F. Cheng, Rate of convergence of Hermite-Fejér polynomials for functions with derivatives of bounded variation, Acta Math. Hungar., 59 (1992), 91-102.  doi: 10.1007/BF00052094.  Google Scholar

[6]

R. Bojanić and M. Vuilleumier, On the rate of convergence of Fourier-Legendre series of functions of bounded variation, J. Approx. Theory, 31 (1981), 67-79.  doi: 10.1016/0021-9045(81)90031-9.  Google Scholar

[7]

F. H. Chêng, On the rate of convergence of Bernstein polynomials of functions of bounded variation, J. Approx. Theory, 39 (1983), 259-274.  doi: 10.1016/0021-9045(83)90098-9.  Google Scholar

[8]

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

[9]

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.  Google Scholar

[10]

F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-203.   Google Scholar

[11]

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

[12]

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

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

A. Lupaş, A q-analogue of the Bernstein operator, in Seminar on Numerical and Statistical Calculus, Univ. "Babeş-Bolyai", Cluj-Napoca, 1987, 85–92.  Google Scholar

[16]

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

[17]

C.-V. Muraru, Note on q-Bernstein-Schurer operators, Stud. Univ. Babeş-Bolyai Math., 56 (2011), 489–495.  Google Scholar

[18]

G. M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math., 4 (1997), 511-518.   Google Scholar

[19]

G. M. Phillips, On generalized Bernstein polynomials, in Numerical Analysis, World Sci. Publ., River Edge, NJ, 1996,263–269. doi: 10.1142/9789812812872_0018.  Google Scholar

[20]

M.-Y. Ren and X.-M. Zeng, On statistical approximation properties of modified q-Bernstein-Schurer operators, Bull. Korean Math. Soc., 50 (2013), 1145-1156.  doi: 10.4134/BKMS.2013.50.4.1145.  Google Scholar

[21]

J. Thomae, Beiträge zur Theorie der durch die Heinische Reihe: Darstellbaren Functionen, J. Reine Angew. Math., 70 (1869), 258-281.  doi: 10.1515/crll.1869.70.258.  Google Scholar

[1]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349

[2]

Tuvi Etzion, Alexander Vardy. On $q$-analogs of Steiner systems and covering designs. Advances in Mathematics of Communications, 2011, 5 (2) : 161-176. doi: 10.3934/amc.2011.5.161

[3]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020210

[4]

Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73

[5]

Xianchao Xiu, Ying Yang, Wanquan Liu, Lingchen Kong, Meijuan Shang. An improved total variation regularized RPCA for moving object detection with dynamic background. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1685-1698. doi: 10.3934/jimo.2019024

[6]

Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53

[7]

Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190

[8]

Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems & Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321

[9]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[10]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192

[11]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008

[12]

Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053

[13]

Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29

[14]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133

[15]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243

[16]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185

[17]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637

[18]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

 Impact Factor: 

Metrics

  • PDF downloads (77)
  • HTML views (175)
  • Cited by (0)

Other articles
by authors

[Back to Top]