基于中心钟形核的极大乘积广义采样算子的饱和性与局部化结果

中文摘要 AI 辅助

本文针对定义在实数集上的非负、有界且一致连续函数，研究了基于合适核函数的极大乘积广义采样算子在一致逼近意义下的饱和阶和局部逆结果。特别地，假设核函数为偶中心钟形函数，我们首先证明$1/w$（$w>0$）是一致饱和阶，相应的饱和类恰为非负常数函数类。这意味着，当在实数集上逼近非平凡（即非常数）的非负、有界且一致连续函数时，$1/w$是极大乘积广义采样算子所能达到的最佳收敛速率。此外，已知对于实数集上的Lipschitz连续函数，当$w \to +\infty$时逼近阶为$1/w$。这里我们证明该结果可以局部反转。具体地，我们证明：若$f$在紧区间$[a,b]\subset\mathbb{R}$上能以$1/w$的速率被逼近，则当$0<a<b$时，$f$在$[a,c]$上对每个$c \in [a,b)$是Lipschitz连续的；当$a<b<0$时，$f$在$[c,b]$上对每个$c \in (a,b]$是Lipschitz连续的。最后，在相同的核假设下，我们针对定义在$[0,1]$上的严格正且有界函数，建立了截断极大乘积广义采样算子序列的强局部化结果。所有这些结果将Coroianu和Gal先前仅针对特定sinc型核建立的结果推广到了更广泛的核函数类。

英文摘要

In this paper, we establish the saturation order and a local inverse result for the uniform approximation of non-negative, bounded, and uniformly continuous functions on $\mathbb{R}$ by max-product generalized sampling operators based on suitable kernel functions. In particular, assuming that the kernel is an even centered bell-shaped function, we first show that $1/w$, $w>0$, is the uniform saturation order, with the corresponding saturation class coinciding with the class of non-negative constant functions. This means that $1/w$ is the best possible rate of convergence that the max-product generalized sampling operators can achieve when approximating non-trivial (i.e., non-constant) non-negative, bounded, and uniformly continuous functions on $\mathbb{R}$. Moreover, it is known that, for Lipschitz continuous functions on $\mathbb{R}$, the approximation order is $1/w$ as $w \to +\infty$. Here, we show that this result can be locally reversed. Specifically, we prove that if $f$ can be approximated at the rate $1/w$ on a compact interval $[a,b]\subset\mathbb{R}$, then $f$ is Lipschitz continuous on $[a,c]$ for every $c \in [a,b)$ whenever $0<a<b$, and on $[c,b]$ for every $c \in (a,b]$ whenever $ a<b<0$. Finally, under the same assumptions on the kernel, we establish a strong localization result for sequences of truncated max-product generalized sampling operators in the case of strictly positive and bounded functions defined on $[0,1]$. All these results extend previous results of Coroianu and Gal, which were established only for specific sinc-type kernels, to a broader class of kernel functions.

发表机构

• University of Perugia（佩鲁贾大学）
• University of Florence（佛罗伦萨大学）

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