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 ...

Let {4>„{dfi)} be a system of orthonormal polynomials on the unit circle with respect to d/i and {y/„,m(dß)} be a system of orthonormal polynomials on the unit circle with respect to the varying measures dß/\wn(z)\2, z = e'e , where {w„(z)} is a sequence of polynomials, degw« = n , whose zeros w„ i, ... , wn,n lie in \z\ < 1 The asymptotic behavior of the ratio of the two systems on and outside...

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...

In this paper we propose using planar and spherical Bernstein polynomials over triangular domain for radiative transfer computations. In the planar domain, we propose using piecewise Bernstein basis functions and symmetric Gaussian quadrature formulas over triangular elements for high quality radiosity solution. In the spherical domain, we propose using piecewise Bernstein basis functions over ...