This paper is concerned with a generalization of the Bernstein polynomials in which the approximated function is evaluated at points spaced in geometric progression instead of the equal spacing of the original polynomials.

It is well known that Bernstein polynomials on triangles preserve monotonicity. In this paper we define and study three kinds of monotonicity preservation of systems of bivariate functions on a triangle. We characterize and compare several of these systems and derive some geometric applications.

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 consider the problem of interpolating scattered data in R 3 by an almost geometrically smooth GC 2 surface, where almost GC 2 means GC 2 except in a finite number of points (the vertices), where the surface is GC 1. A local method is proposed, based on employing so-called degenerate triangular Bernstein-B6zier patches. We give an analysis of quintic patches for GC I and patches of degree ele...

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 ≤ ∞.

We propose a density approximation method based on Bernstein polynomials, consisting in superseding the classical Bernstein operator by a convenient number I∗ of iterates of a closely related operator. We mainly tackle two difficulties met in processing real data, sampled on some mesh XN . The first one consists in determining an optimal sub-mesh XK∗ , in order that the operator associated with...

Abstract — In this paper we derive the matrix of transformation of the Jacobi polynomial basis form into the Bernstein polynomial basis of the same degree n and vice versa. This enables us to combine the superior least-squares performance of the Jacobi polynomials with the geometrical insight of the Bernstein form. Application to the inversion of the Bézier curves is given. 2000 Mathematics Sub...

This article reviews the properties of Tensorial Bernstein Basis (TBB) and its usage, with interval analysis, for solving systems of nonlinear, univariate or multivariate equations resulting from geometric constraints. TBB are routinely used in computerized geometry for geometric modelling in CAD-CAM, or in computer graphics. They provide sharp enclosures of polynomials and their derivatives. T...

Let Q be a real polynomial of degree N all of whose zeros lie in the half-plane Re z < 0. Let M(r, Q) be the maximum of | Q(z) | on \z\= r and n(r,0) the counting function of the zeros of Q. It is shown that the inequality M(r, Q') « (2r)"'{A' + n(r,0)}M(r, Q) holds for r > 0. It is also shown that Bernstein's inequality characterizes polynomials.