一类新的正线性算子保持对数函数
A new class of positive linear operators preserving logarithmic functions
AI总结：
本文提出了一类新的正线性算子，能够再现对数函数，并研究了其收敛性、误差估计及应用
摘要：
在本文中，我们介绍了一类新的正线性算子，该算子推广了经典的伯恩斯坦算子。具体来说，我们构造了一序列算子，能够再现对数函数ln(1+μ+x)，其中μ>0且x∈[0,1]。我们证明了点wise和均匀收敛性，并推导了用连续性模量来表示的近似误差的定量估计。我们还获得了一种Voronovskaja型渐近公式，用于建立饱和结果和逆定理。特别是，所考虑的近似过程的饱和类通过求解二阶微分方程来刻画。此外，还研究了保持形状的性质，如单调性、凹性和变分减少性。最后，简要讨论了一个信号去噪的应用。
英文摘要：
In this paper, we introduce a new class of positive linear operators that generalize the classical Bernstein operators. Specifically, we construct a sequence of operators that reproduce the logarithmic function $\ln(1+μ+x)$, with $μ> 0$ and $x \in [0,1]$. We prove pointwise and uniform convergence and we derive a quantitative estimate of the approximation error in terms of the modulus of continuity. We also obtain a Voronovskaja-type asymptotic formula, that is used to establish saturation results and inverse theorems. In particular, the saturation class of the considered approximation process is characterized by solving a second order differential equation. Shape-preserving properties, such as monotonicity, concavity and variation diminishing, are also investigated. Finally, a simple application to signal denoising is addressed.
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