Bernstein polynomial and discontinuous functions

We obtain new estimates of the rate of convergence of Bernstein operators for discontinuous functions on [0,1]$[0,1]$ which can be used to derive known results for continuous functions and functions of bounded variation.     

Shape preserving representations and optimality of the Bernstein basis

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.Both authors were partially supported by DGICYT PS90-0121.     

Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic

The psi function ψ(x) is defined by ψ(x)=Γ^′(x)/Γ(x), where Γ(x) is the gamma function. We give necessary and sufficient conditions for the function ψ^″(x)+[ψ^′(x+α)]² or its negative to be completely monotonic on (−α,∞), where . We also prove that the function [ψ^′(x)]²+λψ^″(x) is completely monotonic on (0,∞) if and only if λ1. As an application of the latter conclusion, the monotonicity and convexity of the function e^pψ(x+1)−qx with respect to x(−1,∞) are thoroughly discussed for p≠0 and .     

Bernstein Operators for Exponential Polynomials

Let L be a linear differential operator with constant coefficients of order n and complex eigenvalues λ ₀,…,λ _n . Assume that the set U _n of all solutions of the equation Lf=0 is closed under complex conjugation. If the length of the interval [a,b] is smaller than π/M _n , where M _n :=max {|Im λ _j |:j=0,…,n}, then there exists a basis p _n,k , k=0,…,n, of the space U _n with the property that each p _n,k has a zero of order k at a and a zero of order n−k at b, and each p _n,k is positive on the open interval (a,b). Under the additional assumption that λ ₀ and λ ₁ are real and distinct, our first main result states that there exist points a=t ₀<t ₁<⋅⋅⋅<t _n =b and positive numbers α ₀,…,α _n , such that the operator

Bernstein operators for extended Chebyshev systems

Let U_n ⊂ Cⁿ[a, b] be an extended Chebyshev space of dimension n + 1. Suppose that f₀ ∈ U_n is strictly positive and f₁ ∈ U_n has the property that f₁/f₀ is strictly increasing. We search for conditions ensuring the existence of points t₀, …, t_n ∈ [a, b] and positive coefficients α₀, …, α_n such that for all f ∈ C[a, b], the operator B_n:C[a, b] → U_n defined by satisfies B_nf₀ = f₀ and B_nf₁ = f₁. Here it is assumed that p_n,k, k = 0, …, n, is a Bernstein basis, defined by the property that each p_n,k has a zero of order k at a and a zero of order n − k at b.     

On derivatives of Bernstein type rational functions of two variables

In this paper the author defines Bernstein type rational functions of two variables and prove the approximation theorems for the derivatives of them.     

关于Bernstein算子的收敛速度

本文对一类函数建立了Bernstein算子的一致逼近定理,而且给出了其逆定理的一个简短证明。     

Saturation theorems for generalized Bernstein polynomials

In the present paper, we give the explicit formula of the principal part of n ∑ k=0 ([k]q -[n]qx)sxk n-k-1 ∏ m=0 (1-qmx) with respect to [n]q for any integer s and q ∈ (0,1]. And, using the expressions, we obtain saturation theorems for Bn(f,qn;x) approximating to f(x) ∈ C[0,1], 0 < qn ≤ 1, qn → 1.     

Distribution function estimates by Wasserstein metric and Bernstein approximation for C-1 functions

The aim of the paper is to estimate the density functions or distribution functions measured by Wasserstein metric, a typical kind of statistical distances, which is usually required in the statistical learningBased on the classical Bernstein approximation, a scheme is presented.To get the error estimates of the scheme, the problem turns to estimating the L1 norm of the Bernstein approximation for monotone C-1functions, which was rarely discussed in the classical approximation theoryFinally, we get a probability estimate by the statistical distance.     

Weighted approximation of functions on the real line by Bernstein polynomials

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.     

On the degree elevation of Bernstein polynomial representation

The polynomials determined in the Bernstein (Bézier) basis enjoy considerable popularity in computer-aided design (CAD) applications. The common situation in these applications is, that polynomials given in the basis of degree n have to be represented in the basis of higher degree. The corresponding transformation algorithms are called algorithms for degree elevation of Bernstein polynomial representations. These algorithms are only then of practical importance if they do not require the ill-conditioned conversion between the Bernstein and the power basis. We discuss all the algorithms of this kind known in the literature and compare them to the new ones we establish. Some among the latter are better conditioned and not more expensive than the currently used ones. All these algorithms can be applied componentwise to vector-valued polynomial Bézier representations of curves or surfaces.     

Simple bounds for solutions of monotone complementarity problems and convex programs

For a solvable monotone complementarity problem we show that each feasible point which is not a solution of the problem provides simple numerical bounds for some or all components of all solution vectors. Consequently for a solvable differentiable convex program each primal-dual feasible point which is not optimal provides simple bounds for some or all components of all primal-dual solution vectors. We also give an existence result and simple bounds for solutions of monotone compementarity problems satisfying a new, distributed constraint qualification. This result carries over to a simple existence and boundedness result for differentiable convex programs satisfying a similar constraint qualification.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. This material is based on work sponsored by National Science Foundation Grants MCS-8200632 and MCS-8102684.     

Bernstein n-widths for classes of convolution functions with kernels satisfying certain oscillation properties

In this paper, we consider some classes of 2π-periodic convolution functions Bp, and Kp with kernels having certain oscillation properties, which include the classical Sobolev class as special case. With the help of the spectral of nonlinear integral equations, we determine the exact values of Bernstein n-width of the classes Bp, Kp in the space Lp for 1 〈 p 〈 ∞.     

Asymptotic expansion and extrapolation for Bernstein polynomials with applications

Given a real functionf C ^2k [0,1],k 1 and the corresponding Bernstein polynomials {B _n (f)} _n we derive an asymptotic expansion formula forB _n (f). Then, by applying well-known extrapolation algorithms, we obtain new sequences of polynomials which have a faster convergence thanB _n (f). As a subclass of these sequences we recognize the linear combinations of Bernstein polynomials considered by Butzer, Frentiu, and May [2, 6, 9]. In addition we prove approximation theorems which extend previous results of Butzer and May. Finally we consider some applications to numerical differentiation and quadrature and we perform numerical experiments showing the effectiveness of the considered technique.This work was partially supported by a grant from MURST 40.     

On an extremal relation of Bernstein operators

In this note we present a new characterization of Bernstein operators by showing that they are the only solution of a certain extremal relation.     

On the Bernstein Inequality for Rational Functions with a Prescribed Zero

We extend some results of Giroux and Rahman (Trans. Amer. Math. Soc.193(1974), 67–98) for Bernstein-type inequalities on the unit circle for polynomials with a prescribed zero atz=1 to those for rational functions. These results improve the Bernstein-type inequalities for rational functions. The sharpness of these inequalities is also established. Our approach makes use of the Malmquist–Walsh system of orthogonal rational functions on the unit circle associated with the Lebesgue measure.     

Functional calculus of Hermitian elements and Bernstein inequalities

An element a of a unital Banach algebra A is said to be Hermitian if ‖ exp(ita)‖ = 1 for t ∈ ?. We consider some problems concerned with the functional calculus of Hermitian elements and related to estimates for the norm of ?(a), where ? is an admissible function (symbol). Let K be a compact set in ?, and let a be a Hermitian element whose spectrum coincides with K. Then ‖? (a)‖_A ≤ ‖?(D)‖_K, where D is the differential operator ?id/dx and ‖?(D)‖_K is the norm of ?(D) in the Bernstein space B _K of L ^∞(?)-functions whose Fourier transforms are supported in K. We find a differential equation for the extremals of ?(D) and describe them explicitly in the case of an arbitrary complex polynomial ?.     

Bernstein型算子的拓广及其收敛性

取插值基函数Pnk（x，α），导出推广的Bernstein算子Bn（α）（f，x）和推广的Bernstein－Kantorovich算子Kn（α）（f，x），并证明其收敛性．     

New representation of the remainder in the Bernstein approximation

This short note presents a new representation of the remainder in the Bernstein approximation based on divided differences and some immediate applications. It is the only known representation of the remainder in the Bernstein approximation of arbitrary functions as a convex combination of divided differences of second order on known knots. As an application we obtain sharp inequalities for functions possessing bounded divided differences of second order and a new proof of the classical Weierstrass approximation theorem.