A survey of results on the q-Bernstein polynomials

IMA Journal of Numerical Analysis

Volumn 30, Issue 1, 2010, Pages 277-288

  Phillips, George M ^a  

^a UNIVERSITY OF ST ANDREWS   (United Kingdom)

Author keywords

Bernstein polynomials; Convexity; Q Bernstein basis; Q integers; Total positivity

Indexed keywords

RATIONAL FUNCTIONS;

BERNSTEIN BASIS; BERNSTEIN POLYNOMIAL; CONVEXITY; Q-INTEGERS; TOTAL POSITIVITY;

POLYNOMIALS;

EID: 76549126480     PISSN: 02724979     EISSN: 14643642     Source Type: Journal    
DOI: 10.1093/imanum/drn088     Document Type: Article

References (41)

1. AREA, I., GODOY, E., WOZNY, P., LEWANOWICZ, S. & RONVEAUX, A. (2004) Formulae relating little q-Jacobi, q-Hahn and q-Bernstein polynomials: application to q-B?ezier curve evaluation. Integr. Transf. Spec. Funct., 15, 375-385.
2. BERENS, H. & LORENTZ, G. G. (1972) Inverse theorems for Bernstein polynomials. Indiana Univ. Math. J., 21, 693-708.
3. BERNSTEIN, S. N. (1912) Démonstration du théor̀eme de Weierstrass fondée. Commun. Kharkov Math. Soc., 13, 1-2.
4. BOHMAN, H. (1952) On approximation of continuous and of analytic functions. Ark. Mat., 2, 43-56.
5. B̈UŸUKYAZICI, I. (2009) Direct and inverse results for generalized q-Bernstein polynomials. Far East J. Appl. Math. (in press).
6. DAVIS, P. J. (1976) Interpolation and Approximation. New York: Dover.
7. DERRIENNIC, M.-M. (2005) Modified Bernstein polynomials and Jacobi polynomials in q-calculus. Rend. Circ. Mat. Palermo Ser. II, 76 (suppl), 269-290.
8. DIS.IB̈U ŸUK, C. . & ORUC. , H. (2007) A generalization of rational Bernstein-B?ezier curves. BIT, 47, 313-323.
9. DIS.IB̈U ŸUK, C. . & ORUC. , H. (2008) Tensor product q-Bernstein polynomials. BIT, 48, 689-700.
10. DITZIAN, Z. & TOTIK, V. (1987) Moduli of Smoothness. Berlin: Springer.
11. FINTA, Z. (2009) Direct and converse results for q-Bernstein operators. Proc. Edinb. Math. Soc. (in press).
12. GOODMAN, T. N. T., ORUC. , H., & PHILLIPS, G. M. (1999) Convexity and generalized Bernstein polynomials. Proc. Edinb. Math. Soc., 42, 179-190.
13. IL'INSKII, A. & OSTROVSKA, S. (2002) Convergence of generalized Bernstein polynomials. J. Approx. Theory, 116, 100-112.
14. KELISKY, R. P. & RIVLIN, T. J. (1967) Iterates of Bernstein polynomials. Pac. J. Math., 21, 511-520.
15. KOROVKIN, P. P. (1953) On convergence of linear positive operators in the space of continuous functions. Dokl. Akad. Nauk SSSR, 90, 961-964.
16. LEWANOWICZ, S. & WOZ NY, P. (2004) Generalized Bernstein polynomials. BIT, 44, 63-78.
17. LEWANOWICZ, S. & WOZNY, P. (2006) Dual generalized Bernstein basis. J. Approx. Theory, 138, 129-150.
18. LEWANOWICZ, S., WÓZNY, P., AREA, I. & GODOY, E. (2008) Multivariate generalized Bernstein polynomials: identities for orthogonal polynomials of two variables. Numer. Algorithms, 49, 199-220.
19. LUPAS. , A. (1987) A q-analogue of the Bernstein operator. University of Cluj-Napoca, Seminar on Numerical and Statistical Calculus, Preprint No. 9, pp. 85-92.
20. NOWAK, G. (2009) Approximation properties for generalized q-Bernstein polynomials. J. Math. Anal. Appl., 350, 50-55.
21. ORUC. , H. (1998) Generalized Bernstein polynomials and total positivity. Ph.D. Thesis, School of Mathematical and Computational Sciences, University of St Andrews, St. Andrews.
22. ORUC. , H. & PHILLIPS, G. M. (1999) A generalization of the Bernstein polynomials. Proc. Edinb. Math. Soc., 42, 403-413.
23. ORUC. , H. & PHILLIPS, G. M. (2003) q-Bernstein polynomials and B?ezier curves. J. Comput. Appl. Math., 151, 1-12.
24. ORUC., H. & TUNCER, N. (2002) On the convergence and iterates of q-Bernstein polynomials. J. Approx. Theory, 117, 301-313.
25. OSTROVSKA, S. (2003) q-Bernstein polynomials and their iterates. J. Approx. Theory, 123, 232-255.
26. OSTROVSKA, S. (2005) On the q-Bernstein polynomials. Adv. Stud. Contemp. Math., 11, 193-204.
27. OSTROVSKA, S. (2006a) On the improvement of analytic properties under the limit q-Bernstein operators. J. Approx. Theory, 138, 37-53.
28. OSTROVSKA, S. (2006b) On the Lupas. q-analogue of the Bernstein operator. Rocky Mt. J. Math., 36, 1615-1629.
29. OSTROVSKA, S. (2006c) The approximation by q-Bernstein polynomials in the case q. 1. Arch. Math., 86, 282-288.
30. OSTROVSKA, S. (2007) The approximation of logarithmic function by q-Bernstein polynomials in the case q 1. Numer. Algorithms, 44, 69-82.
31. OSTROVSKA, S. (2008) q-Bernstein polynomials of the Cauchy kernel. Appl. Math. Comput., 198, 261-270.
32. PHILLIPS, G. M. (1996a) A de Casteljau algorithm for generalized Bernstein polynomials. BIT, 36, 232-236.
33. PHILLIPS, G. M. (1996b) On generalized Bernstein polynomials. Numerical Analysis: A. R. Mitchell 75th Birthday Volume (D. F. Griffiths & G. A. Watson eds). Singapore: World Scientific, pp. 263-269.
34. PHILLIPS, G. M. (1997) Bernstein polynomials based on the q-integers. The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. Rivlin. Ann. Numer. Math., 14, 511-518.
35. PHILLIPS, G. M. (2003) Interpolation and Approximation by Polynomials. New York: Springer.
36. STANCU, D. D. (1968) Approximation of functions by a new class of linear polynomial operators. Rev. Roum. Math. Pures Appl., 13, 1173-1194.
37. STANCU, D. D. (1983) Approximation of functions by means of a new generalized Bernstein operator. Calcolo, 20, 211-229.
38. STANCU, D. D. (1985) On the representation by divided differences of the remainder in Bernstein's approximation formula. University of Cluj-Napoca, Seminar on Numerical and Statistical Calculus, Preprint No. 4, pp. 103-110.
39. VIDENSKII, V. S. (2005) On some classes of q-parametric positive operators. Oper. Theory Adv. Appl., 158, 213-222.
40. WANG, H. (2005) Korovkin-type theorem and application. J. Approx. Theory, 132, 258-264.
41. WANG, H. (2007) Voronovskaya-type formulas and saturation of convergence for q-Bernstein polynomials for 0 < q < 1. J. Approx. Theory, 145, 182-195