Inegalităţi de tip Chebyshev-Grüss pentru operatorii Bernstein-Euler-Jacobi

Generalized Chebyshev polynomials of the second kind

We characterize the generalized Chebyshev polynomials of the second kind (Chebyshev-II), and then we provide a closed form of the generalized Chebyshev-II polynomials using the Bernstein basis. These polynomials can be used to describe the approximation of continuous functions by Chebyshev interpolation and Chebyshev series and how to efficiently compute such approximations. We conclude the paper with some results concerning integrals of the generalized Chebyshev-II and Bernstein polynomials.

推广的Chebyshev空间上广义Bernstein算子的保形性质

Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

在扩展的Chebyshev空间$U_{n}$的框架下，研究了固定严格正函数$f_{0}$的广义Bernstein算子$B_{n}$的存在性和保形性，以及固定第二个函数$f_{1}$使得$f_{1}/f_{0}$严格增的广义Bernstein算子$B_{n}$的保形性.第一个主要结果给出了存在性的一个归纳准则：假设存在一个结点严格递增的Bernstein算子$B_{n}:C[a,b]\to U_{n}$，固定$f_{0}, f_{1}\in U_{n}$ .如果$U_{n}\subset U_{n + 1}$和$U_{n + 1}$具有非负Bernstein基，则存在具有严格递增节点的Bernstein算子$B_{n+1}:C[a,b]\to U_{n+1}$，固定$f_{0}$和$f_{1}.$。特别地，如果$% f_{0},f_{1},...,f_{n}$是$U_{n}$的基，使得对于每个$k=0,...,n$，$% f_{0},..,f_{k}$的线性跨度是$[ a,b] $上的扩展Chebyshev空间，则存在一个Bernstein算子$B_{n}$，其增加的节点固定$f_{0}$和$f_{1}.$。第二个主要结果表明，在上述假设下，下列不等式成立B_{n} f \geq B_{n+1} f \geq f对于所有$(f_{0},f_{1})$ -凸函数$f\in C[ a,b…

扩展Chebyshev系统的Bernstein算子

Bernstein Operators for Extended Chebyshev Systems

设$U_{n}\subset C^{n}[ a,b] $是维数为$n+1$的扩展切比雪夫空间.设$f_{0}\in U_{n}$是严格正的，且$% f_{1}\in U_{n}$具有$f_{1}/f_{0}$是严格递增的性质.我们寻找确保点$% t_{0},...,t_{n}\in [ a,b] $和正系数$α_{0},...,α_{n}$存在的条件，使得对于所有$f\in C[ a,b]$，由$% B_{n}f=\sum_{k=0}^{n}f(t_{k}) α_{k}p_{n,k}$定义的算子$B_{n}:C[ a,b] \to U_{n}$满足$% B_{n}f_{0}=f_{0}$和$B_{n}f_{1}=f_{1}.$。这里假设$% p_{n,k},k=0,...,n$是Bernstein基，由每个$% p_{n,k}$在$a$处具有阶$k$的零点和在$b.$处具有阶$n-k$的零点的性质定义

广义Bernstein算子的最优性

Optimality of generalized Bernstein operators

我们表明，一定的最优性性质的经典伯恩斯坦运营商也持有，适当重新解释时，广义伯恩斯坦运营商的扩展切比雪夫系统。

Characterization of the generalized Chebyshev-type polynomials of first kind

Orthogonal polynomials have very useful properties in the solution of mathematical problems, so recent years have seen a great deal in the field of approximation theory using orthogonal polynomials. In this paper, we characterize the generalized Chebyshev-type polynomials of the first kind $\mathscr{T}_{n}^{(M,N)}(x),$ then we provide a closed form of the constructed polynomials in term of the Bernstein polynomials $B_{k}^{n}(x).$ We conclude the paper with some results on the integration of the weighted generalized Chebyshev-type with the Bernstein polynomials.

第一类广义切比雪夫Koornwinder型多项式的基础变换

Change of basis for generalized Chebyshev Koornwinder's type polynomials of first kind

本文提供了第一类广义切比雪夫Koornwinder型多项式以固定次数n的伯恩斯坦基表示的显式形式，并研究了其与伯恩斯坦多项式之间的基变换。