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

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

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

Harun Karsli ^,

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 February 2021 Early access 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 $.

Keywords: Q-Bernstein-Schurer operators, pointwise approximation, right and left q-derivatives, convergence rate, bounded variation.

Mathematics Subject Classification: Primary: 41A25; Secondary: 41A35.

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.
[2]	A.-M. Acu, C. V. Muraru, D. 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.
[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.
[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.
[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.
[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.
[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.
[8]	R. J. Finkelstein, q-uncertainty relations, Internat. J. Modern Phys. A., 13 (1998), 1795-1803. doi: 10.1142/S0217751X98000780.
[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.
[10]	F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-203.
[11]	V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0071-7.
[12]	H. Karsli, Some approximation properties of q-Bernstein-Durrmeyer operators, Tbilisi Math. J., 12 (2019), 189-204. doi: 10.32513/tbilisi/1578020576.
[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.
[14]	D. Levi, J. 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.
[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.
[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.
[17]	C.-V. Muraru, Note on q-Bernstein-Schurer operators, Stud. Univ. Babeş-Bolyai Math., 56 (2011), 489–495.
[18]	G. M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math., 4 (1997), 511-518.
[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.
[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.
[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.

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.
[2]	A.-M. Acu, C. V. Muraru, D. 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.
[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.
[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.
[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.
[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.
[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.
[8]	R. J. Finkelstein, q-uncertainty relations, Internat. J. Modern Phys. A., 13 (1998), 1795-1803. doi: 10.1142/S0217751X98000780.
[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.
[10]	F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-203.
[11]	V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0071-7.
[12]	H. Karsli, Some approximation properties of q-Bernstein-Durrmeyer operators, Tbilisi Math. J., 12 (2019), 189-204. doi: 10.32513/tbilisi/1578020576.
[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.
[14]	D. Levi, J. 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.
[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.
[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.
[17]	C.-V. Muraru, Note on q-Bernstein-Schurer operators, Stud. Univ. Babeş-Bolyai Math., 56 (2011), 489–495.
[18]	G. M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math., 4 (1997), 511-518.
[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.
[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.
[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.

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