Recently, Z. Ditzian gave an interesting direct estimate for Bernstein polynomials. In this paper we give direct and inverse results of this type for linear combinations of Bernstein polynomials.

The Bernstein-Bézier representation of polynomials is a very useful tool in computer aided geometric design. In this paper we make use of (multilinear) tensors to describe and manipulate multivariate polynomials in their Bernstein-Bézier form. As application we consider Hermite interpolation with polynomials and splines.

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...

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 ...