We show that the classical Bernstein polynomials BN (f)(x) on the interval [0, 1] (and their higher dimensional generalizations on the simplex Σm ⊂ R) may be expressed in terms of Bergman kernels for the Fubini-Study metric on CP: BN (f)(x) is obtained by applying the Toeplitz operator f(N−1Dθ) to the Fubini-Study Bergman kernels. The expression generalizes immediately to any toric Kähler varie...

Then, for each point x of continuity of f , we have Bn(f )(x)→ f (x) as n→∞. Moreover, if f is continuous on [, ] then Bn(f ) converges uniformly to f as n→∞. Also, for each point x of differentiability of f , we have B′n(f )(x)→ f ′(x) as n→∞ and if f is continuously differentiable on [, ] then B′n(f ) converges to f ′ uniformly as n→∞. Bernstein polynomials have been generalized in the fr...

The introduced method in this study consists of reducing a system of infinite boundary integro-differential equations (IBI-DE) into a system of al- gebraic equations, by expanding the unknown functions, as a series in terms of Laguerre polynomials with unknown coefficients. Properties of these polynomials and operational matrix of integration are rst presented. Finally, two examples illustra...

This paper describes an algorithm that can be used to effectively solve polynomial constraints over finite domains. Such constraints are expressed in terms of inequalities of polynomials with integer coefficients whose variables are assumed to be defined over proper finite domains. The proposed algorithm first reduces each constraint to a canonical form, i.e., a specific form of inequality, the...

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