Research Article
Year 2021,
Volume: 50 Issue: 4, 1111 - 1122, 06.08.2021
Abstract
References
- [1] G.E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.
- [2] S.N. Bernstein, Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités, Communic. Soc. Math. Charkow série 2 13, 1–2, 1912.
- [3] X. Chen, J. Tan, Z. Liu and J. Xie, Approximation of functions by a new family of generalized Bernstein operators, J. Math. Anal. Appl. 450 (1), 244–261, 2017.
- [4] S. Cooper and S. Waldron, The eigenstructure of the Bernstein operator, J. Approx. Theory, 105 (1), 133–165, 2000.
- [5] S. Cooper and S.Waldron, The diagonalisation of the multivariate Bernstein operator, J. Approx. Theory, 117 (1), 103–131, 2002.
- [6] H. Gonska, I. Raşa and E.D. Stˇanilˇa, The eigenstructure of operators linking the Bernstein and the genuine Bernstein-Durrmeyer operators, Mediterr. J. Math. 11 (2), 561–576, 2014.
- [7] H. Gonska, M. Heilmann and I. Raşa, Eigenstructure of the genuine beta operators of Lupaş and Mühlbach, Stud. Univ. Babeş-Bolyai Math 61 (3), 383–388, 2016.
- [8] M. Heilmann and I. Raşa, Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators, Positivity, 21 (3), 897–910, 2017.
- [9] A. II’inskii and S. Ostrovska, Convergence of generalized Bernstein polynomials, J. Approx. Theory, 116 (1), 100–112, 2002.
- [10] U. Itai, On the eigenstructure of the Bernstein kernel, Electron. Trans. Numer. Anal. 25, 431–438, 2006.
- [11] R.P. Kelisky and T.J. Rivlin, Iterates of Bernstein polynomials, Pacific J. Math. 21, 511–520, 1967.
- [12] A. Lupaş, q-Analogue of the Bernstein operator, in: Seminer on Numerical and Statistical Calculus 9, University of Cluj-Napoca.
- [13] I.Ya. Novikov, Asymptotics of the roots of Bernstein polynomials used in the construction of modified Daubechies wavelets, Math. Notes, 71, (1-2), 217–229, 2002.
- [14] S. Ostrovska, q-Bernstein polynomials and their iterates, J. Approx. Theory, 123 (2), 232–255, 2003.
- [15] S. Ostrovska, The first decade of the q-Bernstein polynomials: results and perspectives, J. Math. Anal. Approx. Theory, 2 (1), 35–51, 2007.
- [16] S. Ostrovska and M. Turan, On the eigenvectors of the q-Bernstein operators, Math. Methods Appl. Sci. 37 (4), 562–570, 2014.
- [17] G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math, 4 (1-4), 511–518, 1997.
- [18] C. Qing-Bo and Xu. Xiao-Wei, Shape-preserving properties of a new family of generalized Bernstein operators, J. Inequal. Appl. 2018, 241, 2018.
- [19] V.S. Videnskii, On some classes of q-parametric positive linear operators Selected topics in Complex analysis, Oper. Theory Adv. Appl. 158, 213–222, 2005.
- [20] H. Wang and S. Ostrovska, The norm estimates for the q-Bernstein operator in the case q > 1. Math. Comp. 79, 353–363, 2010.
- [21] S. Wang and C. Zhang, Eigenstructure for binomial operators, Studia Sci. Math. Hungar. 56 (2), 166–176, 2019.
Year 2021,
Volume: 50 Issue: 4, 1111 - 1122, 06.08.2021
Abstract
The eigenvalues and eigenvectors of $(\alpha,q)$-Bernstein operators are unknown and not studied in the literature. As the main result of this article, the eigenvalues and eigenvectors of $(\alpha,q)$-Bernstein operators are obtained. Moreover, we will give the asymptotic behaviour of these eigenvalues and eigenvectors for all $q>0.$ Some eigenvectors for various values of $\alpha$ and $q$ are depicted.
References
- [1] G.E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.
- [2] S.N. Bernstein, Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités, Communic. Soc. Math. Charkow série 2 13, 1–2, 1912.
- [3] X. Chen, J. Tan, Z. Liu and J. Xie, Approximation of functions by a new family of generalized Bernstein operators, J. Math. Anal. Appl. 450 (1), 244–261, 2017.
- [4] S. Cooper and S. Waldron, The eigenstructure of the Bernstein operator, J. Approx. Theory, 105 (1), 133–165, 2000.
- [5] S. Cooper and S.Waldron, The diagonalisation of the multivariate Bernstein operator, J. Approx. Theory, 117 (1), 103–131, 2002.
- [6] H. Gonska, I. Raşa and E.D. Stˇanilˇa, The eigenstructure of operators linking the Bernstein and the genuine Bernstein-Durrmeyer operators, Mediterr. J. Math. 11 (2), 561–576, 2014.
- [7] H. Gonska, M. Heilmann and I. Raşa, Eigenstructure of the genuine beta operators of Lupaş and Mühlbach, Stud. Univ. Babeş-Bolyai Math 61 (3), 383–388, 2016.
- [8] M. Heilmann and I. Raşa, Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators, Positivity, 21 (3), 897–910, 2017.
- [9] A. II’inskii and S. Ostrovska, Convergence of generalized Bernstein polynomials, J. Approx. Theory, 116 (1), 100–112, 2002.
- [10] U. Itai, On the eigenstructure of the Bernstein kernel, Electron. Trans. Numer. Anal. 25, 431–438, 2006.
- [11] R.P. Kelisky and T.J. Rivlin, Iterates of Bernstein polynomials, Pacific J. Math. 21, 511–520, 1967.
- [12] A. Lupaş, q-Analogue of the Bernstein operator, in: Seminer on Numerical and Statistical Calculus 9, University of Cluj-Napoca.
- [13] I.Ya. Novikov, Asymptotics of the roots of Bernstein polynomials used in the construction of modified Daubechies wavelets, Math. Notes, 71, (1-2), 217–229, 2002.
- [14] S. Ostrovska, q-Bernstein polynomials and their iterates, J. Approx. Theory, 123 (2), 232–255, 2003.
- [15] S. Ostrovska, The first decade of the q-Bernstein polynomials: results and perspectives, J. Math. Anal. Approx. Theory, 2 (1), 35–51, 2007.
- [16] S. Ostrovska and M. Turan, On the eigenvectors of the q-Bernstein operators, Math. Methods Appl. Sci. 37 (4), 562–570, 2014.
- [17] G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math, 4 (1-4), 511–518, 1997.
- [18] C. Qing-Bo and Xu. Xiao-Wei, Shape-preserving properties of a new family of generalized Bernstein operators, J. Inequal. Appl. 2018, 241, 2018.
- [19] V.S. Videnskii, On some classes of q-parametric positive linear operators Selected topics in Complex analysis, Oper. Theory Adv. Appl. 158, 213–222, 2005.
- [20] H. Wang and S. Ostrovska, The norm estimates for the q-Bernstein operator in the case q > 1. Math. Comp. 79, 353–363, 2010.
- [21] S. Wang and C. Zhang, Eigenstructure for binomial operators, Studia Sci. Math. Hungar. 56 (2), 166–176, 2019.
There are 21 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Mathematics |
| Authors | |
| Publication Date | August 6, 2021 |
| Published in Issue | Year 2021 Volume: 50 Issue: 4 |