{
  "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 $\\mathbb{R}$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11157",
  "author_total": 2,
  "authors": [
    {
      "name": "Lorenzo Boccali"
    },
    {
      "name": "Gianluca Vinti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/71b1x6fyzc3tp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11157",
  "fulltext_url": null,
  "id": "71b1x6fyzc3tp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11157",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2613|Approximation Theory and Sequence Spaces"
    ]
  ],
  "openalex": "W7212275647",
  "pdf_url": "https://arxiv.org/pdf/2609.11157",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/71b1x6fyzc3tp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11157",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11157"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Saturation and Localization Results for Max-product Generalized Sampling Operators based on Centered Bell-shaped Kernels",
  "title_text": "Saturation and Localization Results for Max-product Generalized Sampling Operators based on Centered Bell-shaped Kernels",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/71b1x6fyzc3tp",
  "venue": "arXiv",
  "versions": []
}