It is well known that in two or more variables Bernstein polynomi-als do not preserve convexity. Here we introduce two variations, one stronger than the classical notion, the other one weaker, which are preserved. Moreover, a weaker suucient condition for the monotony of subsequent Bernstein polynomials is given. linearly independent, in the course of which d has to be greater than or equal to ...

We introduce the local derivatives of a Weyl algebra and prove a theorem of I. N. Bernstein concerning the existence of certain polynomials relating to the action of local derivatives.

Linear combinations of iterates of Bernstein polynomials exponentially converging to the Lagrange interpolating polynomial are given. The results are applied in CAGD to get an exponentially fast weighted progressive iterative approximation technique to fit data with finer and finer precision. AMS subject classifications: 41A25, 41A36

1 Piecewise Polynomials 2 1.1 Barycentric and Bernstein-Bézier Bases . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 B-Spline Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Trimmed Freeform Patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Implicit Algebraic Surface Patches . . . . . . . . . . . . . . . . . . . . . ....

Let be a closed oriented contour on the Riemann sphere. Let E and F be polynomials of degree n + 1, with zeros respectively on the positive and negative sides of . We compute the [n=n] and [n 1=n] Padé denominator at 1 to f (z) = Z 1 z t dt E (t)F (t) : As a consequence, we compute the nth orthogonal polynomial for the weight 1= (EF ). In particular, when is the unit circle, this leads to an ex...

The mathematical modeling of the large deflections for the cantilever beams leads to a nonlinear differential equation with the mixed boundary conditions. Different numerical methods have been implemented by various authors for such problems. In this paper, two novel numerical techniques are investigated for the numerical simulation of the problem. The first is based on a spectral method utiliz...