Multivariate Bernstein operators and redundant systems

Journal of Approximation Theory, 2002

Let B n be the multivariate Bernstein operator of degree n for a simplex in R s : In this paper, we show that B n is diagonalisable with the same eigenvalues as the univariate Bernstein operator, i.e., l ðnÞ k :¼ n! ðn À kÞ! 1 n k ; k ¼ 1; . . . ; n; 1 ¼ l ðnÞ 1 > l ðnÞ 2 > Á Á Á > l ðnÞ n > 0;

Journal of Approximation Theory, 2006

Here we give a simple proof of a new representation for orthogonal polynomials over triangular domains which overcomes the need to make symmetry destroying choices to obtain an orthogonal basis for polynomials of fixed degree by employing redundancy. A formula valid for simplices with Jacobi weights is given, and we exhibit its symmetries by using the Bernstein-Bézier form. From it we obtain the matrix representing the orthogonal projection onto the space of orthogonal polynomials of fixed degree with respect to the Bernstein basis. The entries of this projection matrix are given explicitly by a multivariate analogue of the 3 F 2 hypergeometric function. Along the way we show that a polynomial is a Jacobi polynomial if and only if its Bernstein basis coefficients are a Hahn polynomial. We then discuss the application of these results to surface smoothing problems under linear constraints.

Computers & Mathematics with Applications, 1995

This paper discusses the criteria of convexity, monotonicity, and positivity of Bernstein-B~zier polynomials over simplexes.

Mediterranean Journal of Mathematics, 2019

We consider some connections between the classical sequence of Bernstein polynomials and the Taylor expansion at the point 0 of a C ∞ function f defined on a convex open subset Ω ⊂ R d containing the d-dimensional simplex S d of R d. Under general assumptions, we obtain that the sequence of Bernstein polynomials converges to the Taylor expansion and hence to the function f together with derivatives of every order not only on S d but also on the whole Ω. This result yields extrapolation properties of the classical Bernstein operators and their derivatives. An extension of the Voronovskaja's formula is also stated.

Mediterranean Journal of Mathematics

We study generalizations of the classical Bernstein operators on the polynomial spaces P n [a, b], where instead of fixing 1 and x, we reproduce exactly 1 and a polynomial f 1 , strictly increasing on [a, b]. We prove that for sufficiently large n, there always exist generalized Bernstein operators fixing 1 and f 1. These operators are defined by non-decreasing sequences of nodes precisely when f ′ 1 > 0 on (a, b), but even if f ′ 1 vanishes somewhere inside (a, b), they converge to the identity.

Honam Mathematical Journal, 2011

We prove two identities for multivariate Bernstein polynomials on simplex, which are considered on a pointwise. In this paper, we study good approximations of Bernstein polynomials for every continuous functions on simplex and the higher dimensional q-analogues of Bernstein polynomials on simplex.

Applied Mathematics and Computation, 2011

Explicit formulae for the Bézier coefficients of the constrained dual Bernstein basis polynomials are derived in terms of the Hahn orthogonal polynomials. Using difference properties of the latter polynomials, efficient recursive scheme is obtained to compute these coefficients. Applications of this result to some problems of CAGD is discussed.

The Visual Computer, 2008

We present a method for solving arbitrary systems of N nonlinear polynomials in n variables over an n-dimensional simplicial domain based on polynomial representation in the barycentric Bernstein basis and subdivision. The roots are approximated to arbitrary precision by iteratively constructing a series of smaller bounding simplices. We use geometric subdivision to isolate multiple roots within a simplex. An algorithm implementing this method in rounded interval arithmetic is described and analyzed. We find that when the total order of polynomials is close to the maximum order of each variable, an iteration of this solver algorithm is asymptotically more efficient than the corresponding step in a similar algorithm which relies on polynomial representation in the tensor product Bernstein basis. We also discuss various implementation issues and identify topics for further study.

Journal of Approximation Theory, 2000

The Bernstein operator B n reproduces the linear polynomials, which are therefore eigenfunctions corresponding to the eigenvalue 1. We determine the rest of the eigenstructure of B n . Its eigenvalues are (n) k := n! (n ? k)! 1 n k ; k = 0; 1; : : :; n; and the corresponding monic eigenfunctions p (n) k are polynomials of degree k, which have k simple zeros in 0; 1]. By using an explicit formula, it is shown that p (n) k converges as n ! 1 to a polynomial related to a Jacobi polynomial. Similarly, the dual functionals to p (n) k converge as n ! 1 to measures that we identify. This diagonal form of the Bernstein operator and its limit, the identity (Weierstrass density theorem), is applied to a number of questions. These include the convergence of iterates of the Bernstein operator, and why Lagrange interpolation (at n + 1 equally spaced points) fails to converge for all continuous functions whilst the Bernstein approximants do. We also give the eigenstructure of the Kantorovich operator. Previously, the only member of the Bernstein family for which the eigenfunctions were known explicitly was the Bernstein{Durrmeyer operator, which is self adjoint.

Constructive Approximation, 1992

In this paper we give a complete expansion formula for Bernstein polynomials defined on a s-dimensional simplex. This expansion for a smooth function f represents the Bernstein polynomial B n (f ) as a combination of derivatives of f plus an error term of order O(n -s ).