On the Degree of Multivariate Bernstein Polynomial Operators

On the Degree of Multivariate Bernstein Polynomial Operators yCharles

Let be a d-dimensional simplex with vertices v 0 ; ; v d and B n (f;) denote the n th degree Bernstein polynomial of a continuous function f on. Dahmen and Micchelli 4] proved that B n (f;) B n+1 (f;); n 2 N; for any convex function f on , and it is clear that a necessary and suucient condition for the inequality to become an identity for all n 2 N is that f is an aane polynomial. Let m be the ...

Multivariate Bernstein operators and redundant systems

The Bernstein operator Bn for a simplex in Rd is naturally defined via the Bernstein basis obtained from the barycentric coordinates given by its vertices. Here we consider a generalisation of this basis and the Bernstein operator, which is obtained from generalised barycentric coordinates that are given for more general configurations of points in Rd . We call the associated polynomials a Bern...

On Bernstein Type Inequalities for Complex Polynomial

In this paper, we establish some Bernstein type inequalities for the complex polynomial. Our results constitute generalizations and refinements of some well-known polynomial inequalities.

Derivatives of Multivariate Bernstein Operators and Smoothness with Jacobi Weights

Iterated Bernstein operators for distribution function and density estimation: Balancing between the number of iterations and the polynomial degree

Despite its slow convergence, the use of the Bernstein polynomial approximation is becoming more frequent in Statistics, especially for density estimation of compactly supported probability distributions. This is due to its numerous attractive properties, from both an approximation (uniform shape-preserving approximation, etc.) and a statistical (bona fide estimation, low boundary bias, etc.) p...