We find an explicit formula for the weighted dual functions of the Bernstein polynomials with respect to the Jacobi weight function using the usual inner product in the Hilbert space L[0,1]. We define the weighted dual functionals of the Bernstein polynomials, which are used to find the coefficients in the least squares approximation. 2006 Elsevier Inc. All rights reserved.

In this article, optimal properties of the Bernstein basis of polynomials are revisited. In particular, these include optimal shape preserving properties and optimal stability for the evaluation in computer aided geometric design, minimal conditioning of its collocation matrices and fastest convergence rates of the corresponding iteration approximation property. Recent advances on stable evalua...

A new numerical method, based on the normalized Bernstein polynomials for solving singular integral equations of Abel type is presented here in this paper. We construct an othonormal family { } i n i b 0 = of polynomials of degree n from the th n degree Bernstein polynomials n i B and use them as a basis to approximate the known and unknown functions ) (x f and ) (x φ respectively in the Abel’s...

A criterion of irreducibility for induction products of evaluation modules of type A affine Hecke algebras is given. It is derived from multiplicative properties of the canonical basis of a quantum deformation of the Bernstein-Zelevinsky ring.

The problem of polynomial regression in which the usual monomial basis is replaced by the Bernstein basis is considered. The coefficient matrix A of the overdetermined system to be solved in the least squares sense is then a rectangular Bernstein-Vandermonde matrix. In order to use the method based on the QR decomposition of A, the first stage consists of computing the bidiagonal decomposition ...

In this paper we compare the de Casteljau algorithm with a more efficient algorithm associated to another shape preserving representation. We prove that the Bernstein basis is better conditioned than the basis associated to the more efficient algorithm. Error analysis results are presented. Numerical experiments are also included.

We show that the tensor product B-spline basis and the triangular Bernstein basis are in some sense best conditioned among all nonnegative bases for the spaces of tensor product splines and multivariate polynomials, respectively. We also introduce some new condition numbers which are analogies of component-wise condition numbers for linear systems introduced by Skeel. x

The purpose of this paper is to establish L error estimates, a Bernstein inequality, and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit n-sphere. In particular, the Bernstein inequality estimates L Bessel-potential Sobolev norms of functions in this space in terms of the minimal separation and the L norm of the function ...

We present three linearization methods to over-approximate non-linear multivariate polynomials with convex polyhedra. The first one is based on the substitution of some variables by intervals. The principle of the second linearization technique is to express polynomials in the Bernstein basis and deduce a polyhedron from the Bernstein coefficients. The last method is based on Handelman’s theore...

S. E. Bernstein and T. H. Carr (1996) and P. M. Pexman and S. J. Lupker (1995) suggested that classifiable individual differences in word-naming performance can account for the varied findings on the naming and memory load task (NMLT; K. R. Paap & R. W. Noel, 1991). Bernstein and Carr's technique of testing their explanation by using performance on part of the NMLT to classify participants is p...