Shape criteria of Bernstein-Bézier polynomials over simplexes
Tian-Xiao He1995, Computers & Mathematics with Applications
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
This paper discusses the criteria of convexity, monotonicity, and positivity of Bernstein-B~zier polynomials over simplexes.
Related papers
On the Bernstein–Bézier form of Jacobi polynomials on a simplex
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.
Convexity and generalized Bernstein polynomials
Proceedings of the Edinburgh Mathematical Society, 1999
In a recent generalization of the Bernstein polynomials, the approximated function f is evaluated at points spaced at intervals which are in geometric progression on [0, 1], instead of at equally spaced points. For each positive integer n, this replaces the single polynomial Bnf by a one-parameter family of polynomials , where 0 < q ≤ 1. This paper summarizes briefly the previously known results concerning these generalized Bernstein polynomials and gives new results concerning when f is a monomial. The main results of the paper are obtained by using the concept of total positivity. It is shown that if f is increasing then is increasing, and if f is convex then is convex, generalizing well known results when q = 1. It is also shown that if f is convex then, for any positive integer n This supplements the well known classical result that when f is convex.
Bézier representation of the constrained dual Bernstein polynomials
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.
Some new properties of Generalized Bernstein polynomials
Let Bm(f ) be the Bernstein polynomial of degree m. The generalized Bernstein polynomials
Convexity of Spherical Bernstein-B´ezier Patches and Circular Bernstein-B´ezier Curves
2015
This paper discusses the criteria of convexity of spherical Bernstein-Bézier patches, circular Bernstein-Bézier curves, and homogeneous Bernstein-Bézier polynomials.
A problem of I. Raşa on Bernstein polynomials and convex functions
2016
We present an elementary proof of a conjecture by I. Raşa which is an inequality involving Bernstein basis polynomials and convex functions. It was affirmed in positive very recently by the use of stochastic convex orderings. Moreover, we derive the corresponding results for Mirakyan-Favard-Szász operators and Baskakov operators.
A sharpening of a problem on Bernstein polynomials and convex function and related results
arXiv (Cornell University), 2018
We present a short proof of a conjecture proposed by I. Raşa (2017), which is an inequality involving basic Bernstein polynomials and convex functions. This proof was given in the letter to I. Raşa (2017). The methods of our proof allow us to obtain some extended versions of this inequality as well as other inequalities given by I. Raşa. As a tool we use stochastic convex ordering relations. We propose also some generalizations of the binomial convex concentration inequality. We use it to insert some additional expressions between left and right sides of the Raşa inequalities. is valid for all x, y ∈ [0, 1]. This inequality involving Bernstein basic polynomials and convex functions was stated as an open problems 25 years ago by I. Raşa. During the Conference on Ulam's Type Stability (Rytro, Poland, 2014), Raşa [12] recalled his problem. Inequalities of type (1.1) have important applications. They are useful when studying whether the Bernstein-Schnabl operators preserve convexity (see [3, 4]). Recently, J. Mrowiec, T. Rajba and S. Wąsowicz [10] affirmed the conjecture (1.1) in positive. Their proof makes heavy use of probability theory. As a tool they applied a concept of stochastic convex orderings, as well as the so-called binomial convex concentration inequality. Later, U. Abel [1] gave an elementary proof of (1.1), which was much shorter than that given in [10]. Very recently, A. Komisarski and T. Rajba [6] gave a new, very short proof of (1.1), which is significantly simpler and shorter than that given by U. Abel [1]. As a tool the authors use both stochastic convex orders as well as the usual stochastic order. Let us recall some basic notations and results on stochastic ordering (see [14]). If µ and ν are two probability distributions such that ϕ(x)µ(dx) ≤ ϕ(x)ν(dx) for all convex functions ϕ : R → R,
Representing polynomials by positive linear functions on compact convex polyhedra
Pacific Journal of Mathematics, 1988
Generalized Bernstein polynomials
2004
We introduce polynomials B n i (x; ω|q), depending on two parameters q and ω, which generalize classical Bernstein polynomials, discrete Bernstein polynomials defined by Sablonnière, as well as q-Bernstein polynomials introduced by Phillips. Basic properties of the new polynomials are given. Also, formulas relating B n i (x; ω|q), big q-Jacobi and q-Hahn (or dual q-Hahn) polynomials are presented.
Generalized Bernstein polynomials and symmetric functions
2002
We begin by classifying all solutions of two natural recurrences that Bernstein polynomials satisfy. The first scheme gives a natural characterization of Stancu polynomials. In Section 2, we identify the Bernstein polynomials as coefficients in the generating function for the elementary symmetric functions, which gives a new proof of total positivity for Bernstein polynomials, by identifying the required determinants as Schur functions.
Related papers
A class of inequalities about bernstein polynomial and their applications
Applied Mathematics-A Journal of Chinese Universities, 2006
In this paper a class of new inequalities about Bernstein polynomial is established.
A Note on the Generalized Bernstein Polynomials
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.
Evaluation algorithms for multivariate polynomials in Bernstein–Bézier form
Journal of Approximation Theory, 2006
The evaluation of multivariate polynomials of n variables in Bernstein-Bézier form is considered. A forward error analysis for the corresponding de Casteljau algorithm and the VS algorithm is performed. We also include algorithms that simultaneously evaluate the polynomial and provide "a posteriori" error bounds, without increasing significantly the computational cost. The sharpness of our running error bounds is shown in the case of trivariate polynomials.
Shape preserving representations and optimality of the Bernstein basis
Advances in Computational Mathematics, 1993
This paper gives an affirmative answer to a conjecture given in [10]: the Bernstein basis has optimal shape preserving properties among all normalized totally positive bases for the space of polynomials of degree less than or equal ton over a compact interval. There is also a simple test to recognize normalized totally positive bases (which have good shape preserving properties), and the corresponding corner cutting algorithm to generate the Bézier polygon is also included. Among other properties, it is also proved that the Wronskian matrix of a totally positive basis on an interval [a, ∞) is also totally positive.
Asymptotic expansion formula for Bernstein polynomials defined on a simplex
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 ).
Linear convexity conditions for rectangular and triangular Bernstein-Bézier surfaces
Computer Aided Geometric Design, 1997
The goal of this paper is to derive linear convexity conditions for Bemstein-Brzier surfaces defined on rectangles and triangles. Previously known linear conditions are improved on, in the sense that the new conditions are weaker. Geometric interpretations are provided. © 1997 Elsevier Science B.V.
Conversion formulas for simploidal Bernstein polynomials
2008
In this report, we define simploidal polynomial functions and simploidal Bernstein bases, which are a generalization of the polynomials and Bernstein bases used in simplicial and tensorial Bézier patches. We then provide formulas for converting between simploidal polynomials expressed in various kinds of simploidal Bernstein bases.
A problem of I. Ra\c{s}a on Bernstein polynomials and convex functions
arXiv: Classical Analysis and ODEs, 2016
We present an elementary proof of a conjecture by I. Rasa which is an inequality involving Bernstein basis polynomials and convex functions. It was affirmed in positive very recently by the use of stochastic convex orderings. Moreover, we derive the corresponding results for Mirakyan-Favard-Szasz operators and Baskakov operators.
The Combination of Bernstein Polynomials with Positive Functions Based on a Positive Parameter \(s\)
Communications in Mathematics and Applications
This paper deals with a sequence of the combination of Bernstein polynomials with a positive function τ and based on a parameter s > − 1 2. These polynomials have preserved the functions 1 and τ. First, the convergence theorem for this sequence is studied for a function f ∈ C[0, 1]. Next, the rate of convergence theorem for these polynomials is descript by using the first, second modulus of continuous and Ditzian-Totik modulus of smoothness. Also, the Quantitative Voronovskaja and Grüss-Voronovskaja are obtained. Finally, two numerical examples are given for these polynomials by chosen a test function f ∈ C[0, 1] and two functions for τ to show that the effect of the different values of s and the different chosen functions τ.
Extrapolation Properties of Multivariate Bernstein Polynomials
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.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.