Abstract

In this study we establish some direct connections between arbitrary positive linear operators and their corresponding nonlinear (more exactly sublinear) max-product versions, with respect to uniform and L^p convergence. There are numerous concrete examples of approximation operators, such as Bernstein-type operators, neural network operators, sampling operators and others, where the linear and the max-product versions converge both uniformly. Here, from the quantitative uniform approximation result for an arbitrary sequence of positive linear operators, we deduce by a simple general method a quantitative uniform approximation result for its max-product counterpart. We also establish convergence with respect to the L^p-norm involving the well-known K-functionals, when the supremum of the kernel is bounded from below. Our results cover the cases of bounded and unbounded domains and the case of the Kantorovich variants of the considered operators. Applications to some max-product operators are presented.

摘要

在这项研究中，我们在任意正线性算子与其对应的非线性（更确切地说是次线性）最大乘积版本之间建立了一些直接联系，关于均匀和L ^p收敛。有许多近似算子的具体例子，例如伯恩斯坦型算子、神经网络算子、采样算子等，其中线性和最大乘积版本一致收敛。在这里，从任意正线性算子序列的定量均匀逼近结果，我们通过简单的通用方法推导出其最大乘积对应物的定量均匀逼近结果。我们还建立了关于涉及众所周知的L ^p范数的收敛性K-泛函，当内核的上界是从下界。我们的结果涵盖了有界和无界域的情况以及所考虑运算符的 Kantorovich 变体的情况。介绍了一些最大乘积算子的应用。