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.     

Weighted approximation by Bernstein quasiinterpolants for functions with singularities

We introduce a new type of modified Bernstein quasi-interpolants, which can be used to approximate functions with singularities. We establish direct, inverse, and equivalent theorems of the weighted approximation of this modified quasi-interpolants. Some classical results on approximation of continuous functions are generalized to the weighted approximation of functions with singularities.     

Approximation of Smooth Functions on the Semiaxis by Entire Functions of Bounded Half-Degree

In this paper, we study the pointwise approximation of functions defined on the real semiaxis and having an rth derivative bounded almost everywhere. The approximation is performed by means of entire functions of bounded half-degree, which were introduced by S. N. Bernstein. An asymptotically sharp estimate for pointwise approximation of this class of functions is obtained.     

Bezoutian矩阵的一致逼近形式

借助闭区间上的连续函数可以用Bernstein 多项式一致逼近这一事实,将多项式对所生成的经典Bezoutian 矩阵和Bernstein Bezoutian 矩阵推广到C [0,1]上函数对所对应的情形,给出了 Bezoutian 矩阵一致逼近形式的定义,并且得到如下结论：给出了经典 Bezoutian 矩阵的 Barnett 型分解公式和三角分解公式的一致逼近形式;提供了经典Bezoutian 矩阵和Bernstein Bezoutian 矩阵的一致逼近形式的两类算法;得到了上述两种矩阵的一致逼近形式中元素间的两个恒等关系式。最后,利用数值实例对恒等关系式进行验证,结果表明两类算法是有效的。     

Equi-statistical convergence of positive linear operators

Balcerzak, Dems and Komisarski [M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007) 715-729] have recently introduced the notion of equi-statistical convergence which is stronger than the statistical uniform convergence. In this paper we study its use in the Korovkin-type approximation theory. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work. We also compute the rates of equi-statistical convergence of sequences of positive linear operators. Furthermore, we obtain a Voronovskaya-type theorem in the equi-statistical sense for a sequence of positive linear operators constructed by means of the Bernstein polynomials.     

Bernstein型算子同时逼近误差

该文证明了C[0,1]空间中的函数及其导数可以用Bernstein算子的线性组合同时逼近,得到逼近的正定理与逆定理.同时,也证明了Bernstein算子导数与函数光滑性之间的一个等价关系.该文所获结果沟通了Bernstein算子同时逼近的整体结果与经典的点态结果之间的关系.     

单纯形上的Stancu多项式与最佳多项式逼近

作为Bernstein多项式的推广,本文定义单纯形上的多元Stancu多项式.以最佳多项式逼近为度量,建立Stancu多项式对连续函数的逼近定理与逼近阶估计,给出Stancu多项式的一个逼近逆定理,从而用最佳多项式逼近刻划Stancu多项式的逼近特征.     

Approximation by a sequence of operators of the Srivastava‐Gupta type involving the Brenke polynomials with a certain parameter

We introduce a new sequence of linear positive operators by combining the Brenke polynomials and the Srivastava‐Gupta–type operators defined by Srivastava‐Gupta. obtain the moments of the operators and present some classical and statistical approximation properties by means of Korovkin results. Next, we estimate a global result, which includes the Voronovskaya‐type asymptotic formula, local approximation, error estimation in terms of weighted modulus of continuity, and for functions in a Lipschitz‐type space. Lastly, we estimate the rate of approximation for functions with derivatives of bounded variation.     

MULTIVARIATE WEIGHTED BERNSTEIN-TYPE INEQUALITY AND ITS APPLICATIONS

Bernstein inequality played an important role in approximation theory and Fourier analysis.This article first introduces a general system of functions and the socalled multivariate weighted Bernstein,N...     

THE BERNSTEIN ESTIMATES OF VISCOSITY SOLUTIONS OF LINEAR PARABOLIC EQUATIONS

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Rate of Convergence in Trotter’s Approximation Theorem

We give a quantitative estimate of the convergence in Trotter’s approximation theorem on the convergence of iterates of linear operators to an assigned semigroup. An application is given concerning the classical Bernstein operator on the d-dimensional simplex.      

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。     

On approximation of functions by algebraic polynomials in Hölder spaces

We study approximation of functions by algebraic polynomials in the Hölder spaces corresponding to the generalized Jacobi translation and the Ditzian–Totik moduli of smoothness. By using modifications of the classical moduli of smoothness, we give improvements of the direct and inverse theorems of approximation and prove the criteria of the precise order of decrease of the best approximation in these spaces. Moreover, we obtain strong converse inequalities for some methods of approximation of functions. As an example, we consider approximation by the Durrmeyer–Bernstein polynomial operators.     

Approximation of functions by a new construction of Bernstein‐Chlodowsky operators: Theory and applications

The main motivation of this paper is to provide a generalization of Bernstein‐Chlodowsky type operators which depend on function τ by means of two sequences of functions. The newly defined operators fix the test function set {1, τ, τ²} . Then we present the approximation properties of newly defined operators, such as weighted approximation, degree of approximation and Voronovskaya type theorems. Finally, we present a series of numerical examples demonstrating the effectiveness of this newly defined Bernstein‐Chlodowsky operators for computing function approximation.     

Power Series Kernels

We introduce a class of analytic positive definite multivariate kernels which includes infinite dot product kernels as sometimes used in machine learning, certain new nonlinearly factorizable kernels, and a kernel which is closely related to the Gaussian. Each such kernel reproduces in a certain “native” Hilbert space of multivariate analytic functions. If functions from this space are interpolated in scattered locations by translates of the kernel, we prove spectral convergence rates of the interpolants and all derivatives. By truncation of the power series of the kernel-based interpolants, we constructively generalize the classical Bernstein theorem concerning polynomial approximation of analytic functions to the multivariate case. An application to machine learning algorithms is presented.      

On a conjecture of Mergelyan

A solution is presented to the problem of uniform weighted polynomial approximation on bounded simply connected domains in the complex plane, that is analogous to Mergelyan’s solution to the classical Bernstein problem on the real line.     

单纯形上的q-Stancu多项式的最优逼近阶

构造了单纯形上的多元q-Stancu多项式,它是著名的Bernstein多项式和Stancu多项式的推广.建立该类多项式逼近连续函数的上、下界估计,进而给出其对连续函数的最优逼近阶(饱和阶)及其特征刻画.此外,还研究了该类多项式逼近连续函数的饱和类.     

Extension of Bernstein polynomials to infinite dimensional case

The purpose of this paper is to study some new concrete approximation processes for continuous vector-valued mappings defined on the infinite dimensional cube or on a subset of a real Hilbert space. In both cases these operators are modelled on classical Bernstein polynomials and represent a possible extension to an infinite dimensional setting.The same idea is generalized to obtain from a given approximation process for function defined on a real interval a new approximation process for vector-valued mappings defined on subsets of a real Hilbert space.     

Direct and inverse theorems for Bernstein polynomials with inner singularities

We introduce a new type of Bernstein polynomials, which can be used to approximate the functions with inner singularities. The direct and inverse results of the weighted approximation of this new type of combinations are obtained.     

On Weighted Approximation by Modified Bernstein Operators for Functions with Singularities

Della Vecchia et al. （see [2]） introduced a kind of modified Bernstein operators which can be used to approximate functions with singularities at endpoints on [0,1]. In the present paper, we obtain a kind of pointwise Stechkin-type inequalities for weighted approximation by the modified Bemsetin operators.