Quantitative estimates for the Lupas q-analogue of the Bernstein operator

• Autorzy: Finta Z.
• Języki publikacji: EN

Abstrakty

EN

We establish quantitative results for the approximation properties of the q-analogue of the Bernstein operator defined by Lupas in 1987 and for the approximation properties of the limit Lupas operator introduced by Ostrovska in 2006, via Ditzian-Totik modulus of smoothness. Our results are local and global approximation theorems.

Słowa kluczowe

EN

Bernstein operator; Lupas q-analogue of the Bernstein operator; K-functional; Ditzian-Totik modulus of smoothness

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2011
• Tom: Vol. 44, nr 1
• Strony: 123--130
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Finta Z.

• Babes-Bolyai University Department of Mathematics 1, M. Kogălniceanu st. 400084 Cluj-Napoca, Romania

Bibliografia

• [1] O. Agratini, On certain q-analogues of the Bernstein operators, Carpathian J. Math. 24 (3) (2008), 281–286.
• [2] Z. Ditzian, Direct estimate for Bernstein polynomials, J. Approx. Theory 79 (1994), 165–166.
• [3] Z. Ditzian, V. Totik, Moduli of Smoothness, Springer, New York, 1987.
• [4] G. Freud, On approximation by positive linear methods II, Studia Sci. Math. Hungar. 3 (1968), 365–370.
• [5] A. Il’inskii, S. Ostrovska, Convergence of generalized Bernstein polynomials, J. Approx. Theory 116 (2002), 100–112.
• [6] A. Lupaş, A q-analogue of the Bernstein operator, Babeş-Bolyai University, Seminar on Numerical and Statistical Calculus 9 (1987), 85–92.
• [7] S. Ostrovska, The first decade of the q-Bernstein polynomials: results and perspectives, J. Math. Anal. Approx. Theory 2 (1) (2007), 35–51.
• [8] S. Ostrovska, On the Lupaş q-analogue of the Bernstein operator, Rocky Mountain J. Math. 36 (5) (2006), 1615–1629.
• [9] G. M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997), 511–518.
• [10] T. Popoviciu, Sur l’approximation des functiones convexes d’ordre supér ieure, Mathematica 10 (1935), 49–54.
• [11] V. S. Videnskii, A remark on the rational linear operators considered by A. Lupaş, Some current problems in modern mathematics and education in mathematics, Ross. Gos. Ped. Univ., St. Petersburg, 134–146, 2008 (in Russian).
• [12] H. Wang, Y. Zhang, The rate of convergence for the Lupaş q-analogue of the Bernstein operator, Preprint 08/311, Middle East Technical University, Ankara, Turkey, 2008.