In 1967 Durrmeyer introduced a modiication of the Bernstein polynomials as a selfadjoint polynomial operator on L 2 0; 1] which proved to be an interesting and rich object of investigation. Incorporating Jacobi weights Berens and Xu obtained a more general class of operators, sharing all the advantages of Durrmeyer's modiication, and identiied these operators as de la Vall ee{Poussin means with...

Given p polynomials with coefficients in a commutative unitary integral ring C containing Q, we define the notion of a generic Bernstein-Sato polynomial on an irreducible affine scheme V ⊂ Spec(C). We prove the existence of such a non zero rational polynomial which covers and generalizes previous existing results by H. Biosca. When C is the ring of an algebraic or analytic space, we deduce a st...

In paper [4], transformation matrices mapping the Legendre and Bernstein forms of a polynomial of degree n into each other are derived and examined. In this paper, we derive a matrix of transformation of Chebyshev polynomials of the first kind into Bernstein polynomials and vice versa. We also study the stability of these linear maps and show that the Chebyshev–Bernstein basis conversion is rem...

We study polynomial solutions in the Bézier form of the wave equation in dimensions one and two. We explicitly determine which control points of the Bézier solution at two different times fix the solution.

We present a new method that allows us to get a direct proof of the classical Bernstein asymptotics for the error of the best uniform polynomial approximation of |x| on two symmetric intervals. Note, that in addition, we get asymptotics for the polynomials themselves under a certain renormalization. Also, we solve a problem on asymptotics of the best approximation of sgn (x) on [−1,−a] ∪ [a, 1]...

The authors give error estimates, a Voronovskaya-type relation, strong converse results and saturation for the weighted approximation of functions on the real line with Freud weights by Bernstein-type operators. r 2004 Elsevier Inc. All rights reserved.

Note: This document is meant to be viewed in color. If you only have a black and white version you can download a colored version at http://www.math.utah.edu/~pa/bform.pdf Introduction. There are many ways to write a polynomial. One particular representation is the Bernstein-Bézier form (or B-form for short) of a polynomial whose use has become extremely wide spread during the past two decades ...

A new application of Bernstein–Bezoutian matrices, a type of resultant matrices constructed when the polynomials are given in the Bernstein basis, is presented. In particular, the approach to curve implicitization through Sylvester and Bézout resultant matrices and bivariate interpolation in the usual power basis is extended to the case in which the polynomials appearing in the rational paramet...

We constructmultiple representations relative to different bases of the generalized Tschebyscheff polynomials of second kind. Also, we provide an explicit closed from of The generalized Polynomials of degree r less than or equal n in terms of the Bernstein basis of fixed degree n. In addition, we create the change-of-basis matrices between the generalized Tschebyscheff of the second kind polyno...