{
  "abstract": "One of the basic problems in mathematical learning theory is to identify a function $f: Ω\\to {\\mathbb R}$ with certain specific properties which fits a given training data $\\{(x_i, ξ_i)\\in Ω\\times {\\mathbb R}: i=1, \\ldots, n\\}$, in the sense that, $f(x_i) = ξ_i$ for $i=1, \\ldots, n$, where $Ω$ is a compact subset of ${\\mathbb R}^d$ for some $d\\in {\\mathbb N}$ and $f$ is required to have certain specified characteristics. We address this problem when $f$ belongs to an arbitrary normed linear space ${X}$, and the evaluation maps $f\\mapsto f(x_i)$ are replaced by maps of the form $ f\\mapsto φ_i(f)$ on $X$, where $φ_1, \\ldots, φ_n$ are continuous linear functionals on ${X}$. Using an inner product structure on ${\\mathbb R}^n$, we shall device a method of least-squares for obtaining an approximate solution for the above problem and also identify a subspace of ${X}$ in which the least-square solution is unique, which in turn is shown to be equivalent to solving a matrix equation. In the context when the matrix under consideration is ill-conditioned, a regularized equation is devised, and order optimal error estimates are derived for the exact targeted data $ξ=(ξ_1, \\ldots, ξ_n)$ and also when it is noisy, by choosing the regularization parameter appropriately.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11960",
  "author_total": 1,
  "authors": [
    {
      "name": "M. Thamban Nair"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4fqt8jwhn7wc9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11960",
  "fulltext_url": null,
  "id": "4fqt8jwhn7wc9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11960",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2610|Numerical methods in inverse problems"
    ]
  ],
  "openalex": "W7212696972",
  "pdf_url": "https://arxiv.org/pdf/2609.11960",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4fqt8jwhn7wc9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.11960",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11960"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A Finite Data Based Regularization in a Normed Linear Space Setting",
  "title_text": "A Finite Data Based Regularization in a Normed Linear Space Setting",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/4fqt8jwhn7wc9",
  "venue": "arXiv",
  "versions": []
}