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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 |
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 [
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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