In this paper we discuss a natural way to deene barycentric coordinates on general sphere-like surfaces. This leads to a theory of Bernstein-B ezier polynomials which parallels the familiar planar case. Our constructions are based on a study of homogeneous polynomials on trihedra in IR 3. The special case of Bernstein-B ezier polynomials on a sphere is considered in detail.

the objective of this paper is applying the well-known exact operational matrices (eoms) idea for solving the emden-fowler equations, illustrating the superiority of eoms over ordinary operational matrices (ooms). up to now, a few studies have been conducted on eoms ; but the solved differential equations did not have high-degree nonlinearity and the reported results could not strongly show the...

We characterize the generalized Chebyshev polynomials of the second kind (Chebyshev-II), and then we provide a closed form of the generalized Chebyshev-II polynomials using the Bernstein basis. These polynomials can be used to describe the approximation of continuous functions by Chebyshev interpolation and Chebyshev series and how to efficiently compute such approximations. We conclude the pap...

In many applications of interval computations, it turned out to be beneficial to represent polynomials on a given interval [x, x] as linear combinations of Bernstein polynomials (x − x) · (x − x)n−k. In this paper, we provide a theoretical explanation for this empirical success: namely, we show that under reasonable optimality criteria, Bernstein polynomials can be uniquely determined from the ...

Bernstein polynomials on a simplex V are considered. The expansion of a given polynomial p into these polynomials provides bounds for the range of p over V . Bounds for the range of a rational function over V can easily obtained from the Bernstein expansions of the numerator and denominator polynomials of this function. In this paper it is shown that these bounds converge monotonically and line...

Let Bm(f) be the Bernstein polynomial of degree m. The Generalized Bernstein polynomials

and Applied Analysis 3 see 8 . For 0 ≤ k ≤ n, derivatives of the nth degree modified q-Bernstein polynomials are polynomials of degree n − 1: d dx Bk,n ( x, q ) n ( qBk−1,n−1 ( x, q ) − q1−xBk,n−1 ( x, q )) ln q q − 1 1.9 see 8 . The Bernstein polynomials can also be defined in many different ways. Thus, recently, many applications of these polynomials have been looked for by many authors. In t...

The present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schrödinger equations whose Hamiltonian is given by the generalized Laguerre operator. More precisely, we show that ...

A new class of orthogonal polynomials is introduced which generalizes the Bernstein-Szegö polynomials and includes the associated polynomials as well. The purpose of this paper is to give a natural extension of the Bernstein-Szegö orthogonal polynomials for a general class of weight functions. A nonnegative function w defined on the real line is called a weight function if w > 0, fRw > 0 and al...