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Abstract and Applied Analysis
Volume 2011, Article ID 609431, 13 pages
http://dx.doi.org/10.1155/2011/609431
Research Article

A de Casteljau Algorithm for -Bernstein-Stancu Polynomials

The Great Poland University of Social and Economics in Środa Wielkopolska, Paderewskiego 27, 63-000 Środa Wielkopolska, Poland

Received 17 September 2010; Accepted 7 January 2011

Academic Editor: Wolfgang Ruess

Copyright © 2011 Grzegorz Nowak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with a generalization of the -Bernstein polynomials and Stancu operators, where the function is evaluated at intervals which are in geometric progression. It is shown that these polynomials can be generated by a de Casteljau algorithm, which is a generalization of that relating to the classical case and -Bernstein case.

1. Introduction

Let . For any fixed real number and for , the -integers of the number are defined by The -factorial , for , is defined by For the integers , , (), the -binomial or the Gaussian coefficients are defined by (see [1, page 12]) For , , and each positive integer , we introduce (see [2]) the following generalized -Bernstein operators: where Note, that an empty product in (1.5) denotes 1. In the case where , reduces to the well-known -Bernstein polynomials introduced by Phillips [3, 4] in 1997 In the case where , reduces to Bernstein-Stancu polynomials, introduced by Stancu [5] in 1968 When and , we obtain the classical Bernstein polynomial defined by Basic facts on Bernstein polynomials, their generalizations, and applications can be found for example in [68]. In recent years, the -Bernstein polynomials have attracted much interest, and a great number of interesting results related to the polynomials have been obtained (see [3, 4, 912]). Some approximation properties of the Stancu operators are presented in [5, 1315].

Let , for , and recursively, for and . It is easily established by induction that -differences satisfy the relation In [2], we prove that the operators defined by (1.4) can be expressed in terms of -differences which generalized the well-known result [3, 4] for the -Bernstein polynomial. In this paper, we show that polynomials defined by (1.4) can be generated by a de Castljau algorithm, which is a generalization of that relating to the classical case [16] and -Bernstein case [4, 11].

2. Auxiliary Results

We note that defined by (1.4), is a monotone linear operator for any and . These operators reproduces linear functions [2], that is, They also satisfy the end point interpolation conditions and . These properties are significant in designing curves and surfaces.

Moreover, the following holds.

Lemma 2.1. Let , . Then, for all , and .

Proof. We use induction on . First, we see from equality , , that (2.2) is evident for . Let us assume that (2.2) holds for a given . Then, using (2.2), we obtain where Using the obvious equalities we have It is easy to see that Therefore, From last equality and (2.3), we obtain This completes the proof of the lemma.

3. Main Result

The generalized -Bernstein polynomials, defined by (1.4), may be evaluated by Algorithm 1.

alg1
Algorithm 1: De Casteljau type algorithm.

In the case, where , this is the de Casteljau algorithm for evaluating the -Bernstein polynomial [3, 4]. Note that with and , we recover the original classical de Casteljau algorithm (see Hoschek and Lasser [16]). The algorithm is justifed by the following theorem.

Theorem 3.1. Each intermediate point of the algorithm can be expressed as and, in particular

Proof. We use induction on . From the initial conditions in the algorithm, , , it is clear that (3.1) holds for and . Let us assume that (3.1) holds for some such that , and for all such that . Then, for , it follows from the algorithm that and using (3.1), we obtain We see that and hence, It is easy to verify that Therefore, Consequently, Thus, one has the desired result.

Theorem 3.2. For and , we have for all .

Proof. Using (2.2) and (3.1), we have where First, we prove that for all , , and . Note that an empty sum denotes 0.
We use the induction on . First, we see that (3.13) holds for and all . Let us assume that (3.13) holds for a given , and for all . Then, from (3.12) and (3.13), we obtain We see that and hence, Next, in view of the equality we obtain (3.13). Consequently, in view of (3.11) and (3.13), we get Next, in view of the equality we obtain The condition (1.10) completes the proof.

Theorems 3.1 and 3.2 are generalizations of Theorems 2.1 and 2.3 in [11].

Note that when and , (3.10) does indeed reduce to (1.11)

References

  1. V. Kac and P. Cheung, Quantum Calculus, Springer, New York, NY, USA, 2002.
  2. G. Nowak, “Approximation properties for generalized q-Bernstein polynomials,” Journal of Mathematical Analysis and Applications, vol. 350, no. 1, pp. 50–55, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  3. G. M. Phillips, “Bernstein polynomials based on the q-integers,” Annals of Numerical Mathematics, vol. 4, no. 1–4, pp. 511–518, 1997. View at Google Scholar
  4. G. M. Phillips, Interpolation and Approximation by Polynomials, Springer, New York, NY, USA, 2003.
  5. D. D. Stancu, “Approximation of functions by a new class of linear polynomial operators,” Académie de la République Populaire Roumaine. Revue Roumaine de Mathématiques Pures et Appliquées, vol. 13, pp. 1173–1194, 1968. View at Google Scholar · View at Zentralblatt MATH
  6. G. G. Lorentz, Bernstein Polynomials, vol. 8, University of Toronto Press, Toronto, Canada, 1953.
  7. P. Pych-Taberska, “Some approximation properties of Bernšteĭn and Kantorovič polynomials,” Functiones et Approximatio Commentarii Mathematici, vol. 6, pp. 57–67, 1978. View at Google Scholar · View at Zentralblatt MATH
  8. P. Pych-Taberska, “On the rate of pointwise convergence of Bernstein and Kantorovič polynomials,” Functiones et Approximatio Commentarii Mathematici, vol. 16, pp. 63–76, 1988. View at Google Scholar · View at Zentralblatt MATH
  9. T. N. T. Goodman, H. Oruç, and G. M. Phillips, “Convexity and generalized Bernstein polynomials,” Proceedings of the Edinburgh Mathematical Society. Series 2, vol. 42, no. 1, pp. 179–190, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. A. II'inskii and S. Ostrovska, “Convergence of generalized Bernstein polynomials,” Journal of Approximation Theory, vol. 116, no. 1, pp. 100–112, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. G. M. Phillips, “A de Casteljau algorithm for generalized Bernstein polynomials,” BIT. Numerical Mathematics, vol. 37, no. 1, pp. 232–236, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. G. M. Phillips, “A survey of results on the q-Bernstein polynomials,” IMA Journal of Numerical Analysis, vol. 30, no. 1, pp. 277–288, 2010. View at Publisher · View at Google Scholar
  13. Z. Finta, “Direct and converse results for Stancu operator,” Periodica Mathematica Hungarica, vol. 44, no. 1, pp. 1–6, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Z. Finta, “On approximation properties of Stancu's operators,” Mathematica, vol. 47, no. 4, pp. 47–55, 2002. View at Google Scholar
  15. H. H. Gonska and J. Meier, “Quantitative theorems on approximation by Bernstein-Stancu operators,” Calcolo, vol. 21, no. 4, pp. 317–335, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. J. Hoschek and D. Lasser, Fundamentals of Computer Aided Geometric Design, A. K. Peters, Wellesley, Mass, USA, 1993.