Saturation and Localization Results for Max-product Generalized Sampling Operators based on Centered Bell-shaped Kernels
Abstract
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 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 ℝ. Moreover, it is known that, for Lipschitz continuous functions on ℝ, the approximation order is 1/w as w → +∞. 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]⊂ℝ, then f is Lipschitz continuous on [a,c] for every c ∈ [a,b) whenever 0<a<b, and on [c,b] for every c ∈ (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.
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BibTeX
@misc{boccali2026saturation,
title = {{Saturation and Localization Results for Max-product Generalized Sampling Operators based on Centered Bell-shaped Kernels}},
author = {Lorenzo Boccali and Gianluca Vinti},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11157},
eprint = {2609.11157},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11157},
}
RIS
TY - UNPB
TI - Saturation and Localization Results for Max-product Generalized Sampling Operators based on Centered Bell-shaped Kernels
AU - Lorenzo Boccali
AU - Gianluca Vinti
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11157
UR - https://arxiv.org/abs/2609.11157
AB - 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 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 ℝ. Moreover, it is known that, for Lipschitz continuous functions on ℝ, the approximation order is 1/w as w → +∞. 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]⊂ℝ, then f is Lipschitz continuous on [a,c] for every c ∈ [a,b) whenever 0<a<b, and on [c,b] for every c ∈ (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.
ER -