A de Casteljau Algorithm for q -Bernstein-Stancu Polynomials

This paper is concerned with a generalization of the q -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 q -Bernstein case.


Introduction
Let q > 0. For any fixed real number q > 0 and for n ∈ Z {0, ±1, ±2, . . .}, the q-integers of the number n are defined by n 1 − q n 1 − q , for q / 1, n n, for q 1.

2 Abstract and Applied Analysis
For the integers n, k, n ≥ k ≥ 0 , the q-binomial or the Gaussian coefficients are defined by see 1, page 12 For f ∈ C 0; 1 , q > 0, α ≥ 0 and each positive integer n, we introduce see 2 the following generalized q-Bernstein operators: Note, that an empty product in 1.5 denotes 1. In the case where α 0, B q,α n f; x reduces to the well-known q-Bernstein polynomials introduced by Phillips 3, 4 in 1997 In the case where q 1, B q,α n f; x reduces to Bernstein-Stancu polynomials, introduced by Stancu 5 in 1968 When q 1 and α 0, we obtain the classical Bernstein polynomial defined by Basic facts on Bernstein polynomials, their generalizations, and applications can be found for example in 6-8 . In recent years, the q-Bernstein polynomials have attracted much interest, and a great number of interesting results related to the B n,q f polynomials have been obtained see 3, 4, 9-12 . Some approximation properties of the Stancu operators are presented in 5, 13-15 .
Let Δ 0 q f j f j , for j 0, 1, . . . , n, and recursively, Abstract and Applied Analysis 3 for k 0, 1, . . . , n − j − 1 and f j f j / n . It is easily established by induction that qdifferences satisfy the relation In 2 , we prove that the operators B q,α n f; x defined by 1.4 can be expressed in terms of q-differences which generalized the well-known result 3, 4 for the q-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 q-Bernstein case 4, 11 .

Auxiliary Results
We note that B q,α n f; x defined by 1.4 , is a monotone linear operator for any 0 < q ≤ 1 and α ≥ 0. These operators reproduces linear functions 2 , that is, They also satisfy the end point interpolation conditions B q,α n f; 0 f 0 and B q,α n f; 1 f 1 . These properties are significant in designing curves and surfaces.
Moreover, the following holds.
Proof. We use induction on m. First, we see from equality −r −q −r r , r ∈ N , that 2.2 is evident for m 1. Let us assume that 2.2 holds for a given m ∈ N. Then, using 2.2 , we obtain

2.6
It is easy to see that s .

2.9
This completes the proof of the lemma.

Main Result
The generalized q-Bernstein polynomials, defined by 1.4 , may be evaluated by Algorithm 1.
In the case, where α 0, this is the de Casteljau algorithm for evaluating the q-Bernstein polynomial 3, 4 . Note that with q 1 and α 0, we recover the original classical de Casteljau algorithm see Hoschek and Lasser 16 . The algorithm is justifed by the following theorem. 3.2 Proof. We use induction on m. From the initial conditions in the algorithm, f 0 r f r / n f r , 0 ≤ r ≤ n, it is clear that 3.1 holds for m 0 and 0 ≤ r ≤ n. Let us assume that 3.1 holds for some m such that 0 ≤ m < n, and for all r such that 0 ≤ r ≤ n−m. Then, for 0 ≤ r ≤ n−m−1, it follows from the algorithm that  x α r s .

8 Abstract and Applied Analysis
We see that and hence, x α r s .

3.6
It is easy to verify that

3.12
First, we prove that for all m ∈ N 0 {0, 1, 2, . . .}, t ∈ N 0 , and x ∈ 0; 1 . Note that an empty sum denotes 0. We use the induction on m. First, we see that 3.13 holds for m 0 and all t ∈ N 0 . Let us assume that 3.13 holds for a given m, and for all t ∈ N 0 . Then, from 3.12 and 3.13 , we obtain

3.14
We see that