We define the notion of Bernstein measures and Bernstein approximations over general convex polytopes. This generalizes well-known Bernstein polynomials which are used to prove the Weierstrass approximation theorem on one dimensional intervals. We discuss some properties of Bernstein measures and approximations, and prove an asymptotic expansion of the Bernstein approximations for smooth functi...

We begin by classifying all solutions of two natural recurrences that Bernstein polynomials satisfy. The first scheme gives a natural characterization of Stancu polynomials. In Section 2, we identify the Bernstein polynomials as coefficients in the generating function for the elementary symmetric functions, which gives a new proof of total positivity for Bernstein polynomials, by identifying th...

Global and local Bernstein-Sato ideals, Bernstein-Sato polynomials and Bernstein-Sato polynomials of varieties are introduced, their basic properties are proven and their algorithmic determination with the method of Briançon/Maisonobe is presented. Strati cations with respect to the local variants of the introduced polynomials and ideals with the methods of Bahloul/Oaku and Levandovskyy/Martín-...

in this paper, the ritz-galerkin method in bernstein polynomial basis is applied for solving the nonlinear problem of the magnetohydrodynamic (mhd) flow of third grade fluid between the two plates. the properties of the bernstein  polynomials together with the ritz-galerkin method are used to reduce the solution of the mhd couette flow of non-newtonian fluid in a porous medium to the solution o...

We present approximation kernels for orthogonal expansions with respect to Bernstein-Szegö polynomials. The construction is derived from known results for Chebyshev polynomials of the first kind and does not pose any restrictions on the Bernstein-Szegö polynomials.

In this paper, the Ritz-Galerkin method in Bernstein polynomial basis is applied for solving the nonlinear problem of the magnetohydrodynamic (MHD) flow of third grade fluid between the two plates. The properties of the Bernstein  polynomials together with the Ritz-Galerkin method are used to reduce the solution of the MHD Couette flow of non-Newtonian fluid in a porous medium to the solution o...

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...

Let S be a polynomial of degree 2n + 2, that is, positive on the real axis, and let w = 1/S on (−∞,∞). We present an explicit formula for the nth orthogonal polynomial and related quantities for the weight w. This is an analogue for the real line of the classical Bernstein-Szegő formula for (−1, 1). 1. The result The Bernstein-Szegő formula provides an explicit formula for orthogonal polynomial...