In [20] a generalization of Bernstein polynomials and Bézier curves based on umbral calculus has been introduced. In the present paper we describe new geometric and algorithmic properties of this generalization including: (1) families of polynomials introduced by Stancu [19] and Goldman [12], i.e., families that include both Bernstein and Lagrange polynomial, are generalized in a new way, (2) a...

They investigated pointwise convergence properties of (1) in a compact sub-interval of [0,∞). Then Gadjiev and Çakar [2] obtained uniform convergence of (1) on semi-axis [0,∞) on some subspace of bounded and continuous functions by using the test functions ( x 1+x) ν , ν = 0, 1, 2. In 1996 q-based generalization of the classical Bernstein polynomials were introduced by G. M. Phillips [3]. He ha...

In this paper we propose an efficient numerical technique for solving fractional initial value problems. It is based on the Bernstein polynomials. We derive an explicit form for the Bernstein operational matrix of fractional order integration. Numerical results are presented. In order to show the efficiency of the presented method, we compare our results with some operational matrix techniques.

We show that the size of the 1-norm condition number of the univariate Bernstein basis for polynomials of degree n is O(2n/ √ n). This is consistent with known estimates [3], [5] for p = 2 and p = ∞ and leads to asymptotically correct results for the p-norm condition number of the Bernstein basis for any p with 1 ≤ p ≤ ∞.

Fast and efficient methods of evaluation of the connection coefficients between shifted Jacobi and Bernstein polynomials are proposed. The complexity of the algorithms is O(n), where n denotes the degree of the Bernstein basis. Given results can be helpful in a computer aided geometric design, e.g., in the optimization of some methods of the degree reduction of Bézier curves.

We establish a …rst order asymptotic for the entropy integrals Z I pn log pn W 2 and Z I pn log (pnW ) 2 W 2 where fpngn=0 are the orthonormal polynomials associated with the exponential weight W . 1 The Result Let I = (c; d) be a real interval, where 1 c < 0 < d 1; and let Q : I ! [0;1) be convex. Let W := exp ( Q) and assume that all power moments Z I xW (x)dx; n = 0; 1; 2; 3; ::: are …nite. ...

Bernstein polynomials are a classical tool in Computer Aided Design to create smooth maps with a high degree of local control. They are used for the construction of Bézier surfaces, free-form deformations, and many other applications. However, classical Bernstein polynomials are only defined for simplices and parallelepipeds. These can in general not directly capture the shape of arbitrary obje...

We combine recently-developed finite element algorithms based on Bernstein polynomials [1, 14] with the explicit basis construction of the finite element exterior calculus [5] to give a family of algorithms for the Rham complex on simplices that achieves stiffness matrix construction and matrix-free action in optimal complexity. These algorithms are based on realizing the exterior calculus base...