Strong converse inequalities for the weighted multivariate Bernstein-Durrmeyer operator on the simplex via multipliers

Generalized Bernstein-Durrmeyer Operators of Blending Type

In this article we present the Durrmeyer variant of generalized Bernstein operators that preserve the constant functions involving non-negative parameter ?. We derive the approximation behaviour of these operators including global approximation theorem via Ditzian-Totik modulus of continuity, the order of convergence for the Lipschitz type space. Furthermore, we study a Voronovskaja type asymptotic formula and local approximation theorem by means of second order modulus of smoothness. Furthermore, we obtain the rate of approximation for absolutely continuous functions having a derivative equiv…

Quantitative estimates in approximation by Bernstein-Durrmeyer-Choquet operators with respect to monotone and submodular set functions

For the qualitative results of pointwise and uniform approximation obtained in \cite{Gal-Opris}, we present general quantitative estimates in terms of the modulus of continuity and in terms of a $K$-functional, for the generalized multivariate Bernstein-Durrmeyer operator $M_{n, Γ_{n, x}}$, written in terms of the Choquet integral with respect to a family of monotone and submodular set functions, $Γ_{n, x}$, on the standard $d$-dimensional simplex. When $Γ_{n, x}$ reduces to two elements, one a Choquet submodular set function and the other one a Borel measure, for suitable modified Bernstein-D…

正线性算子的复合核

Kernels for composition of positive linear operators

本文研究了 Bernstein--Durrmeyer 算子与 Szász--Mirakjan--Durrmeyer 算子的复合，建立了复合核的新恒等式并证明了交换性，同时给出了迭代算子的线性组合表示。

Power series of the operators $U_n^{\varrho}$

We study power series of members of a class of positive linear operators reproducing linear function and constituting a link between genuine Bernstein-Durrmeyer and classical Bernstein operators. Using the eigenstructure of the operators we give a non-quantitative convergence result towards the inverse Voronovskaya operators. We include a quantitative statement via a smoothing approach.

Lagrange-type operators associated with $U_n^{\varrho}$

We consider a class of positive linear operators which, among others, constitute a link between the classical Bernstein operators and the genuine Bernstein-Durrmeyer mappings. The focus is on their relation to certain Lagrange-type interpolators associated to them, a well known feature in the theory of Bernstein operators. Considerations concerning iterated Boolean sums and the derivatives of the operator images are included. Our main tool is the eigenstructure of the members of the class.

(p,q)-Beta Functions and Applications in Approximation

In the present paper, we consider (p,q)-analogue of the Beta operators and using it, we propose the integral modification of the generalized Bernstein polynomials. We estimate some direct results on local and global approximation. Also, we illustrate some graphs for the convergence of (p,q)-Bernstein-Durrmeyer operators for different values of the parameters p and q using Mathematica package.

Geometric series of positive linear operators and inverse Voronovskaya theorem

We define the associated geometric series for a large class of positive linear operators and study the convergence of the series in the case of sequences of admissible operators. We obtain an inverse Voronovskaya theorem and we apply our results to the Bernstein operators, the Bernstein-Durrmeyer-type operators, and the symmetrical version of Meyer-König and Zeller operators.

单纯形上的尖锐Bernstein不等式

Sharp Bernstein inequalities on simplex

本文在单纯形上建立了两类新的Bernstein不等式：一类基于谱算子得到Jacobi权$L^2$范数下的尖锐不等式，另一类针对倍增权给出$L^p$范数不等式，并指出当$d\ge3$时前者不一定是后者的特例。

正线性算子及其导数差值的估计

Estimates for the differences of positive linear operators and their derivatives

本文研究了特定正线性算子及其导数差值的估计问题，采用伯恩斯坦算子、 Kantorovich算子、真实伯恩斯坦-杜尔梅耶算子、带有雅可比权的杜尔梅耶算子等定义在有界区间上的算子，以一阶连续性模为量化形式给出估计，并通过数值例子分析理论结果。

A comparative analysis of Bernstein type estimates for the derivative of multivariate polynomials

We compare the yields of two methods to obtain Bernstein type pointwise estimates for the derivative of a multivariate polynomial in points of some domain, where the polynomial is assumed to have sup norm at most 1. One method, due to Sarantopoulos, relies on inscribing ellipses into the convex domain K. The other, pluripotential theoretic approach, mainly due to Baran, works for even more general sets, and yields estimates through the use of the pluricomplex Green function (the Zaharjuta -Siciak extremal function). Using the inscribed ellipse method on non-symmetric convex domains, a key role…