The sharpness of convergence results for $q$-Bernstein polynomials in the case $q>1$

Ostrovska, Sofiya

Czechoslovak Mathematical Journal, Tome 58 (2008) no. 4, pp. 1195-1206

Abstract (VO)

Due to the fact that in the case $q>1$ the $q$-Bernstein polynomials are no longer positive linear operators on $C[0,1],$ the study of their convergence properties turns out to be essentially more difficult than that for $q1.$ In this paper, new saturation theorems related to the convergence of $q$-Bernstein polynomials in the case $q>1$ are proved.

Détail

Classification : 30E10, 33D45, 41A10
Keywords: $q$-integers; $q$-binomial coefficients; $q$-Bernstein polynomials; uniform convergence; analytic function; Cauchy estimates

Bibliographie

[1] Cao, J.-D.: A generalization of the Bernstein polynomials. J. Math. Anal. Appl. 209 (1997), 140-146. | DOI | MR | Zbl

[2] Cao, J.-D., Gonska, H., Kacsó, D.: On the impossibility of certain lower estimates for sequences of linear operators. Math. Balkanica 19 (2005), 39-58. | MR

[3] Cheney, E. W., Sharma, A.: On a generalization of Bernstein polynomials. Riv. Mat. Univ. Parma 5 (1964), 77-84. | MR | Zbl

[4] Cooper, S., Waldron, S.: The eigenstructure of the Bernstein operator. J. Approx. Theory 105 (2000), 133-165. | DOI | MR | Zbl

[5] Derriennic, M.-M.: Modified Bernstein polynomials and Jacobi polynomials in $q$-calculus. Rendiconti Del Circolo Matematico Di Palermo, Serie II, Suppl. 76 (2005), 269-290. | MR | Zbl

[6] Gonska, H.: The rate of convergence of bounded linear processes on spaces of continuous functions. Automat. Comput. Appl. Math. 7 (1999), 38-97. | MR

[7] Gupta, V.: Some approximation properties of $q$-Durrmeyer operators. Appl. Math. Comput. 197 (2008), 172-178. | DOI | MR | Zbl

[8] Gupta, V., Wang, H.: The rate of convergence of $q$-Durrmeyer operators for $0. Math. Meth. Appl. Sci. (2008). MR 2447215

[9] Habib, A., Umar, S.: On generalized Bernstein polynomials. Indian J. Pure Appl. Math. 11 (1980), 177-189. | MR | Zbl

[10] Il'inskii, A., Ostrovska, S.: Convergence of generalized Bernstein polynomials. J. Approx. Theory 116 (2002), 100-112. | DOI | MR | Zbl

[11] Lorentz, G. G.: Bernstein Polynomials. Chelsea, New York (1986). | MR | Zbl

[12] Lupaş, A.: A $q$-analogue of the Bernstein operator. University of Cluj-Napoca, Seminar on numerical and statistical calculus, No. 9 (1987).

[13] Novikov, I. Ya.: Asymptotics of the roots of Bernstein polynomials used in the construction of modified Daubechies wavelets. Mathematical Notes 71 (2002), 217-229. | DOI | MR | Zbl

[14] Ostrovska, S.: $q$-Bernstein polynomials and their iterates. J. Approx. Theory 123 (2003), 232-255. | DOI | MR | Zbl

[15] Ostrovska, S.: On the $q$-Bernstein polynomials. Advanced Studies in Contemporary Mathematics 11 (2005), 193-204. | MR | Zbl

[16] Ostrovska, S.: On the improvement of analytic properties under the limit $q$-Bernstein operator. J. Approx. Theory 138 (2006), 37-53. | DOI | MR | Zbl

[17] Petrone, S.: Random Bernstein polynomials. Scan. J. Statist. 26 (1999), 373-393. | DOI | MR | Zbl

[18] Videnskii, V. S.: On some classes of $q$-parametric positive operators. Operator Theory, Advances and Applications, Vol. 158 (2005), 213-222. | DOI | MR

[19] Videnskii, V. S.: On the polynomials with respect to the generalized Bernstein basis. In: Problems of modern mathematics and mathematical education, Hertzen readings. St.-Petersburg (2005), 130-134 Russian.

[20] Wang, H.: Korovkin-type theorem and application. J. Approx. Theory 132 (2005), 258-264. | DOI | MR | Zbl

[21] Wang, H.: Voronovskaya type formulas and saturation of convergence for $q$-Bernstein polynomials for $0. J. Approx. Theory 145 (2007), 182-195. DOI 10.1016/j.jat.2006.08.005 | MR 2312464 | Zbl 1112.41016

[22] Wang, H., Wu, X. Z.: Saturation of convergence for $q$-Bernstein polynomials in the case $q\geq 1$. J. Math. Anal.Appl. 337 (2008), 744-750. | DOI | MR

[23] Wang, H.: Properties of convergence for $\omega,q$-Bernstein polynomials. J. Math. Anal. Appl. 340 (2008), 1096-1108. | DOI | MR | Zbl