Abstract:
In this paper, a de Casteljau algorithm to compute (p,q)-Bernstein Bezier curves based on (p,q)-integers is introduced. We study the nature of degree elevation and degree reduction for (p,q)-Bezier Bernstein functions. The new curves have some properties similar to q-Bezier curves. Moreover, we construct the corresponding tensor product surfaces over the rectangular domain (u, v) \in [0, 1] \times [0, 1] depending on four parameters. We also study the de Casteljau algorithm and degree evaluation properties of the surfaces for these generalization over the rectangular domain. Furthermore, some fundamental properties for (p,q)-Bernstein Bezier curves are discussed. We get q-Bezier curves and surfaces for (u, v) \in [0, 1] \times [0, 1] when we set the parameter p1 = p2 = 1.

Abstract:
In this paper, we use the blending functions of Bernstein polynomials with shifted knots for construction of Bezier curves and surfaces. We study the nature of degree elevation and degree reduction for Bezier Bernstein functions with shifted knots. Parametric curves are represented using these modified Bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve. We get Bezier curve defined on [0, 1] when we set the parameter \alpha=\beta to the value 0. We also present a de Casteljau algorithm to compute Bernstein Bezier curves and surfaces with shifted knots. The new curves have some properties similar to Bezier curves. Furthermore, some fundamental properties for Bernstein Bezier curves and surfaces are discussed.

Abstract:
We investigate shape preserving for -Bernstein-Stancu polynomials introduced by Nowak in 2009. When , reduces to the well-known -Bernstein polynomials introduced by Phillips in 1997; when , reduces to Bernstein-Stancu polynomials introduced by Stancu in 1968; when , , we obtain classical Bernstein polynomials. We prove that basic basis is a normalized totally positive basis on and -Bernstein-Stancu operators are variation-diminishing, monotonicity preserving and convexity preserving on . 1. Introduction Let . For each nonnegative integer , we define the -integer as we then define -factorial as and we next define a -binomial coefficient as for integers and as zero otherwise. Also, we use the -Pochhammer symbol defined as for any For , , , and each positive integer , we will investigate the following -Bernstein-Stancu operator introduced by Nowak in 2009 [1]: where Note that empty product in (6) denotes 1. In this case, when , reduces to the well-known -Bernstein polynomials introduced by Phillips [2] in 1997: When , reduces to Bernstein-Stancu polynomials introduced by Stancu [3] in 1968: When and , we obtain the classical Bernstein polynomials defined by Now, we review and state some general properties of -Bernstein-Stancu operators. It follows directly from the definition that -Bernstein-Stancu operators possess the endpoint interpolation property, that is, and leave invariant linear function: They are also degree reducing on polynomials; that is, if is a polynomial of degree , then is a polynomials of degree ≤ . Taking , in (11), we conclude that In 2009, Nowak proved that the -Bernstein-Stancu operators can be expressed in terms of -differences [1]: where At the same time, he still showed that, for , , For a real-valued function on an interval , we define to be the number of sign changes of ; that is, where the supremum is taken over all increasing sequence in , for all . We say that is variation-diminishing if Similarly, for a matrix , we say is variation-diminishing if, for any vector for which is defined, then . Let be a sequence of positive linear operators on . We say that is monotonicity preserving if is increasing (decreasing) for an increasing (decreasing) function on . We say that is convexity preserving if is convex (concave) for a convex (concave) function on . Let , and let be the sequence of basic -Bernstein-Stancu polynomials, and denote by the sequence of all polynomials of degree at most ; then is a basis for (see [1]). Hence, there exists a nonsingular transformation matrix from to such that A matrix is said to be totally positivity

Abstract:
In this paper, we use the blending functions of Lupas type (rational) (p,q)-Bernstein operators based on (p,q)-integers for construction of Lupas (p,q)-Beezier curves (rational curves) and surfaces (rational surfaces) with two shape parameters. We study the nature of degree elevation and degree reduction for Lupas (p,q)-Bezier Bernstein functions. Parametric curves are represented using Lupas (p,q)-Bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve. We get q-Bezier curve when we set the parameter p to the value 1: We also introduce a de Casteljau algorithm for Lupas type (p,q)-Bernstein Bezier curves. The new curves have some properties similar to q-Bezier curves. Moreover, we construct the corresponding tensor product surfaces over the rectangular domain (u,v) \in [0,1] \times [0,1] depending on four parameters. We also study the de Casteljau algorithm and degree evaluation properties of the surfaces for these generalization over the rectangular domain. Furthermore, some fundamental properties for Lupas type (p,q)-Bernstein Bezier curves are discussed. We get q-Bezier curves and surfaces for (u,v) \in [0,1] \times [0,1] when we set the parameter p1 = p2 = 1. In Comparison to q-Bezier curves and surfaces based on Phillips q-Bernstein polynomials, our generalizations show more flexibility in choosing the value of p1; p2 and q1; q2 and superiority in shape control of curves and surfaces. The shape parameters provide more convenience for the curve and surface modeling.

Abstract:
As a generalization of the Bernstein polynomials, the q-Bernstein polynomialsand the Stancu polynomials, a class of broader polynomials called the multivariateq-Stancu polynomials defined on the normal simplex is introduced. With the partial continuity modulus and the full continuity modulus of multivariate function as a metric, the uniform convergence theorem of the generalized multivariate Stancu polynomials to any continuous functions is proved, and the estimation of the convergence order is also obtained. Finally, an example shows the validity of the obtained result.

Abstract:
研究了一种新近引入的修正Durrmeyer型Bernstein-Stancu算子在[0，1]区间上的逼近性质，建立了点态逼近的正、逆定理. Abstract：Recently, DONG et al introduced a new kind of Durrmeyer type Bernstein-Stancu operators, and investigated the approximation properties of the new operators on a subset of[0,1]. In this paper, we obtained both the direct and converse results of the approximation by the operators on the whole interval[0,1].

Abstract:
In the present paper, we introduce the Chlodowsky variant of (p,q) Bernstein-Stancu-Schurer operators which is a generalization of (p,q) Bernstein-Stancu-Schurer operators. We have also discussed its approximation properties and rate of convergence.

Abstract:
In this paper, we introduce a Kantorovich type generalization of q-Bernstein-Stancu operators. We study the convergence of the introduced operators and also obtain the rate of convergence by these operators in terms of the modulus of continuity. Further, we study local approximation property and Voronovskaja type theorem for the said operators. We show comparisons and some illustrative graphics for the convergence of operators to a certain function.

Abstract:
Positive polynomial operator that approximates Urison operator, when integration domain is a "regular triangle" is investigated. We obtain Bernstein Polynomials as a particular case.

Abstract:
A new recursive algorithm for efficient computation of the B\'ezier coefficients of dual bivariate Bernstein polynomials is given. Applications of this result to some approximation problems of Computer Aided Geometric Design (CAGD) are discussed.