The paper introduces q-parametric Bleimann, Butzer and Hahn (q-BBH) operators as a rational transformation of q-Bernstein-Lupaş operators. On their basis, a set of new results on q-BBH operators can be obtained easily from the corresponding properties of q-Bernstein-Lupaş operators. Furthermore, convergence properties of q-BBH operators are studied.

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.

The q-Bernstein basis with 0 < q < 1 emerges as an extension of the Bernstein basis corresponding to a stochastic process generalizing Bernoulli trials forming a totally positive system on [0, 1]. In the case q > 1, the behavior of the q-Bernstein basic polynomials on [0, 1] combines the fast increase in magnitude with sign oscillations. This seriously complicates the study of qBernstein polyno...

The q-Bernstein basis with 0 < q < 1 emerges as an extension of the Bernstein basis corresponding to a stochastic process generalizing Bernoulli trials forming a totally positive system on [0, 1]. In the case q > 1, the behavior of the q-Bernstein basic polynomials on [0, 1] combines the fast increase in magnitude with sign oscillations. This seriously complicates the study of qBernstein polyno...

The Bernstein-Bézier form of a polynomial is widely used in the fields of computer aided geometric design, spline approximation theory and, more recently, for high order finite element methods for the solution of partial differential equations. However, if one wishes to compute the classical Lagrange interpolant relative to the Bernstein basis, then the resulting Bernstein-Vandermonde matrix is...

A numerical approach for solving constant-coefficient differential equations whose solutions exhibit boundary layer structure is built by inserting Bernstein Partition of Unity into Galerkin variational weak form. Due to the reproduction capability of Bernstein basis, such implementation shows excellent accuracy at boundaries and is able to capture sharp gradients of the field variable by p-ref...

In this paper, we consider the issue of dual functions for the Bernstein basis which satisfy boundary conditions. The Jacobi weight function with the usual inner product in the Hilbert space are used. Some examples of the transformation matrices are given. Some figures for the weighted dual functions of the Bernstein basis with respect to the Jacobi weight function satisfying boundary condition...

Recently, for the sake of fitting scattered data points, an important method based on the PIA (progressive iterative approximation) property of the univariate NTP (normalized totally positive) bases has been effectively adopted. We extend this property to the bivariate Bernstein basis over a triangle domain for constructing triangular Bézier surfaces, and prove that this good property is satisf...