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Year 2019, Volume: 68 Issue: 2, 2170 - 2197, 01.08.2019
https://doi.org/10.31801/cfsuasmas.545460

Abstract

References

  • Acu, A.M., Stancu-Schurer-Kantorovich operators based on q-integers, Appl. Math. and Comput., 259, (2015), 896-907.
  • Acu, AM, Sofonea, F, Barbosu, D., Note on a q-analogue of Stancu-Kantorovich operators, Miskolc Mathematical Notes, 16(1), (2015), 3-15.
  • Acu, A.M., Manav, N., Sofonea, D.F., Approximation properties of λ-Kantorovich operators, J. Inequal. Appl., (2018), 2018:202.
  • Acu, A.M., Muraru, C.V., Sofonea, D.F., Radu, V.A.:, Some approximation properties of a Durrmeyer variant of q-Bernstein-Schurer operators, Mathematical Methods in Applied Science, 39(18), (2016), 5636-5650.
  • Acu, A.M., Acar, T., Muraru, C.V., Radu, V.A., Some approximation properties by a class of bivariate operators, Mathematical Methods in the Applied Sciences, 42 (2019), 1-15, https://doi.org/10.1002/mma.5515
  • Altomare, F., Cappelletti, M. M. and Leonessa, V., On a generalization of Szasz-Mirakjan-Kantorovich operators, Results Math. 63(3-4), (2013), 837-863.
  • Aral, A., Gupta, V. and Agarwal, R. P., Applications of q-Calculus in Operator Theory, Springer, 2013.
  • Badea, C., Badea, I. and Gonska, H. H., Notes on the degree of approximation of B-continuous and B-differentiable functions, J. Approx. Theory Appl. 4, (1988), 95-108.
  • Başcanbaz-Tunca, G., Erençin, A., Ince-Ilarslan, H.G., Bivariate Cheney-Sharma operators on simplex, Hacettepe Journal of Mathematics and Statistics, 47(4), (2018), 793-804.
  • Bögel, K., Mehrdimensionale Differentiation von Funtionen mehrerer Veränderlicher, J. Reine Angew. Math. 170, (1934), 197-217.
  • Bögel, K., Über die mehrdimensionale differentiation, Jahresber. Deutsch. Math.-Verein., 65 (1962), 45-71.
  • Bögel, K., Über die mehrdimensionale differentiation, integration und beschränkte variation, J. Reine Angew. Math., 173, (1935), 5-29
  • Butzer, P.L., Berens, H., Semi-groups of Operators and Approximation, Springer, New York, 1967.
  • Ditzian, Z. and Totik, V., Moduli of smoothness, volume 9 of Springer Series in Computational Mathematics, SpringerVerlag, New York, 1987.
  • Gadjiev, A. D., Theorems of the type of P. P. Korovkin's theorems, Mat. Zametki 20 (5) (1976), 781-786. ((in Russian), Math. Notes 20(5-6) (1976), 995-998 (Engl. Trans.)).
  • Gonska, H., Heilmann M. and Rasa I., Kantorovich operators of order k, Numer. Funct. Anal. Optim., 32(7), (2011), 717-738.
  • Gupta, V., Rassias, T. M., Agarwal, P.M. and Acu, A. M., Recent Advances in Constructive Approximation Theory, Springer, Cham, 2018.
  • Kajla, A., Agrawal, P. N., Szász-Kantorovich type operators based on Charlier polynomials, Kyungpook Math. J., 56(3), (2016), 877-897.
  • Kajla, A., Goyal, M., Modified Bernstein-Kantorovich operators for functions of one and two variables, Rendiconti del Circolo Matematico di Palermo, (2017), DOI10.1007/s12215-017-0320-z.
  • Lupas, A., A q-analogue of the Bernstein operators, Semianr on Numerical and Statistical Calculus, University of Cluj-Napoca, 9 (1987), 85-98.
  • Lupaş, A., q-Analogues of Stancu operators, Mathematical Analysis and Approximation Theory, Burg Verlag , 2002, 145-154.
  • Marchaud, A., Sur les Derivées et sur les Différences des Fonctions de Variables Réclles, J. Math. Pures et Appl. 6 (1927), 337-426.
  • Muraru, C.V., Acu, A.M., Radu, V.A., On the monotonicity of q-Schurer-Stancu type polynomials, Miskolc Mathematical Notes, 19(1), (2018), 19-28.
  • Özarslan, M. A., Duman, O., Smoothnesss properties of modified Bernstein-kantorovich operators, Numer. Func. Anal. Opt. 37 (1) (2016), 92-105.
  • Phillips, G. M., Bernstein polynomials based on q-integers, Ann. Numer. Math., 4 (1997), 511-518.
  • Shisha, O. and Mond, P., The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. USA 60, (1968), 1196-1200.
  • Sucu, S., Ibikli, E., Approximation by means of Kantorovich-Stancu type operators, Numer. Funct. Anal. Optim., 34(5), (2013), 557-575.
  • Thomae, J., Beiträge zue Theorie der durch die Heineshe Reihe., J. Reine Angew Math. 70, (1869), 258-281.

Approximation properties of modified q-Bernstein-Kantorovich operators

Year 2019, Volume: 68 Issue: 2, 2170 - 2197, 01.08.2019
https://doi.org/10.31801/cfsuasmas.545460

Abstract

In the present paper we define a q-analogue of the modified Bernstein-Kantorovich operators
introduced by Ozarslan and Duman (Numer. Funct. Anal. Optim. 37:92-105,2016). We establish
the shape preserving properties of these operators e.g. monotonicity and convexity and study the rate
of convergence by means of Lipschitz class and Peetre's K-functional and degree of approximation with
the aid of a smoothing process e.g Steklov mean. Further, we introduce the bivariate case of modified
q-Bernstein-Kantorovich operators and study the degree of approximation in terms of the partial and
total modulus of continuity and Peetre's K-functional. Finally, we introduce the associated GBS (Generalized
Boolean Sum) operators and investigate the approximation of the Bogel continuous and Bogel
differentiable functions by using the mixed modulus of smoothness and Lipschitz class.

References

  • Acu, A.M., Stancu-Schurer-Kantorovich operators based on q-integers, Appl. Math. and Comput., 259, (2015), 896-907.
  • Acu, AM, Sofonea, F, Barbosu, D., Note on a q-analogue of Stancu-Kantorovich operators, Miskolc Mathematical Notes, 16(1), (2015), 3-15.
  • Acu, A.M., Manav, N., Sofonea, D.F., Approximation properties of λ-Kantorovich operators, J. Inequal. Appl., (2018), 2018:202.
  • Acu, A.M., Muraru, C.V., Sofonea, D.F., Radu, V.A.:, Some approximation properties of a Durrmeyer variant of q-Bernstein-Schurer operators, Mathematical Methods in Applied Science, 39(18), (2016), 5636-5650.
  • Acu, A.M., Acar, T., Muraru, C.V., Radu, V.A., Some approximation properties by a class of bivariate operators, Mathematical Methods in the Applied Sciences, 42 (2019), 1-15, https://doi.org/10.1002/mma.5515
  • Altomare, F., Cappelletti, M. M. and Leonessa, V., On a generalization of Szasz-Mirakjan-Kantorovich operators, Results Math. 63(3-4), (2013), 837-863.
  • Aral, A., Gupta, V. and Agarwal, R. P., Applications of q-Calculus in Operator Theory, Springer, 2013.
  • Badea, C., Badea, I. and Gonska, H. H., Notes on the degree of approximation of B-continuous and B-differentiable functions, J. Approx. Theory Appl. 4, (1988), 95-108.
  • Başcanbaz-Tunca, G., Erençin, A., Ince-Ilarslan, H.G., Bivariate Cheney-Sharma operators on simplex, Hacettepe Journal of Mathematics and Statistics, 47(4), (2018), 793-804.
  • Bögel, K., Mehrdimensionale Differentiation von Funtionen mehrerer Veränderlicher, J. Reine Angew. Math. 170, (1934), 197-217.
  • Bögel, K., Über die mehrdimensionale differentiation, Jahresber. Deutsch. Math.-Verein., 65 (1962), 45-71.
  • Bögel, K., Über die mehrdimensionale differentiation, integration und beschränkte variation, J. Reine Angew. Math., 173, (1935), 5-29
  • Butzer, P.L., Berens, H., Semi-groups of Operators and Approximation, Springer, New York, 1967.
  • Ditzian, Z. and Totik, V., Moduli of smoothness, volume 9 of Springer Series in Computational Mathematics, SpringerVerlag, New York, 1987.
  • Gadjiev, A. D., Theorems of the type of P. P. Korovkin's theorems, Mat. Zametki 20 (5) (1976), 781-786. ((in Russian), Math. Notes 20(5-6) (1976), 995-998 (Engl. Trans.)).
  • Gonska, H., Heilmann M. and Rasa I., Kantorovich operators of order k, Numer. Funct. Anal. Optim., 32(7), (2011), 717-738.
  • Gupta, V., Rassias, T. M., Agarwal, P.M. and Acu, A. M., Recent Advances in Constructive Approximation Theory, Springer, Cham, 2018.
  • Kajla, A., Agrawal, P. N., Szász-Kantorovich type operators based on Charlier polynomials, Kyungpook Math. J., 56(3), (2016), 877-897.
  • Kajla, A., Goyal, M., Modified Bernstein-Kantorovich operators for functions of one and two variables, Rendiconti del Circolo Matematico di Palermo, (2017), DOI10.1007/s12215-017-0320-z.
  • Lupas, A., A q-analogue of the Bernstein operators, Semianr on Numerical and Statistical Calculus, University of Cluj-Napoca, 9 (1987), 85-98.
  • Lupaş, A., q-Analogues of Stancu operators, Mathematical Analysis and Approximation Theory, Burg Verlag , 2002, 145-154.
  • Marchaud, A., Sur les Derivées et sur les Différences des Fonctions de Variables Réclles, J. Math. Pures et Appl. 6 (1927), 337-426.
  • Muraru, C.V., Acu, A.M., Radu, V.A., On the monotonicity of q-Schurer-Stancu type polynomials, Miskolc Mathematical Notes, 19(1), (2018), 19-28.
  • Özarslan, M. A., Duman, O., Smoothnesss properties of modified Bernstein-kantorovich operators, Numer. Func. Anal. Opt. 37 (1) (2016), 92-105.
  • Phillips, G. M., Bernstein polynomials based on q-integers, Ann. Numer. Math., 4 (1997), 511-518.
  • Shisha, O. and Mond, P., The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. USA 60, (1968), 1196-1200.
  • Sucu, S., Ibikli, E., Approximation by means of Kantorovich-Stancu type operators, Numer. Funct. Anal. Optim., 34(5), (2013), 557-575.
  • Thomae, J., Beiträge zue Theorie der durch die Heineshe Reihe., J. Reine Angew Math. 70, (1869), 258-281.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Review Articles
Authors

Ana Maria Acu 0000-0003-1192-2281

Purshottam Agrawal 0000-0003-1192-2281

Dharmendra Kumar This is me 0000-0003-1192-2281

Publication Date August 1, 2019
Submission Date March 27, 2019
Acceptance Date May 31, 2019
Published in Issue Year 2019 Volume: 68 Issue: 2

Cite

APA Acu, A. M., Agrawal, P., & Kumar, D. (2019). Approximation properties of modified q-Bernstein-Kantorovich operators. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(2), 2170-2197. https://doi.org/10.31801/cfsuasmas.545460
AMA Acu AM, Agrawal P, Kumar D. Approximation properties of modified q-Bernstein-Kantorovich operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. August 2019;68(2):2170-2197. doi:10.31801/cfsuasmas.545460
Chicago Acu, Ana Maria, Purshottam Agrawal, and Dharmendra Kumar. “Approximation Properties of Modified Q-Bernstein-Kantorovich Operators”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 2 (August 2019): 2170-97. https://doi.org/10.31801/cfsuasmas.545460.
EndNote Acu AM, Agrawal P, Kumar D (August 1, 2019) Approximation properties of modified q-Bernstein-Kantorovich operators. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 2 2170–2197.
IEEE A. M. Acu, P. Agrawal, and D. Kumar, “Approximation properties of modified q-Bernstein-Kantorovich operators”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 2, pp. 2170–2197, 2019, doi: 10.31801/cfsuasmas.545460.
ISNAD Acu, Ana Maria et al. “Approximation Properties of Modified Q-Bernstein-Kantorovich Operators”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/2 (August 2019), 2170-2197. https://doi.org/10.31801/cfsuasmas.545460.
JAMA Acu AM, Agrawal P, Kumar D. Approximation properties of modified q-Bernstein-Kantorovich operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:2170–2197.
MLA Acu, Ana Maria et al. “Approximation Properties of Modified Q-Bernstein-Kantorovich Operators”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 2, 2019, pp. 2170-97, doi:10.31801/cfsuasmas.545460.
Vancouver Acu AM, Agrawal P, Kumar D. Approximation properties of modified q-Bernstein-Kantorovich operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(2):2170-97.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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