{
  "abstract": "We prove a general lower bound for the Lebesgue constants $\\Lambda_n$ of Leja sequences for every nonpolar compact set \\(K\\subset\\mathbb C\\): $\\textstyle \\limsup_{n \\to+\\infty} \\Lambda_n \\, \\sqrt{n^{-1} W_n(K)}\\geq \\tfrac{1}{\\sqrt{2}}$, where \\(W_n(K)\\) denotes the Chebyshev--Widom factor. In particular, bounded Widom factors imply a universal square root obstruction. We also establish improved polynomial upper bounds on compact subsets $K$ of the real line, building on the approach of Andrievskii and Nazarov. Our first ingredient is a refinement of their Key Lemma: we lower its exponent from $9/8 = 1.125$ to a near-optimal $\\gamma^\\star \\approx 0.6304$. Our second ingredient is a localisation of the separation argument: instead of using the global minimum spacing among the first $n$ Leja points, we retain the local Green-function scale. For finite unions of intervals, a packing argument and sharp estimates on the latter function near and away from the endpoints yield $\\Lambda_n = O(n^{1 + \\gamma^\\star})$, uniformly over Leja sequences. This improves the previously known best exponent $13/4 = 3.25$ to around $1.6304$ for such sets (including the benchmark case $K = [-1,1]$), hence substantially narrowing the gap with the near-linear growth observed numerically. Corresponding polynomial bounds for $\\tau$-Leja sequences are derived.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.01836",
  "author_total": 1,
  "authors": [
    {
      "name": "Camille Pouchol"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4ezs9y8aw6ka1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.01836",
  "fulltext_url": null,
  "id": "4ezs9y8aw6ka1",
  "issns": [],
  "landing_url": "https://hal.science/hal-05676129v2",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
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      "math.CA"
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      "openalex",
      "2604|Mathematical functions and polynomials"
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  "openalex": "W7167210012",
  "pdf_url": "https://hal.science/hal-05676129/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4ezs9y8aw6ka1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:29Z",
      "native_id": "hal-05676129",
      "source": "hal",
      "url": "https://hal.science/hal-05676129v2"
    },
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2607.01836",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.01836"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Lebesgue constants of Leja sequences: a universal lower bound and improved upper bounds on the real line.",
  "title_text": "Lebesgue constants of Leja sequences: a universal lower bound and improved upper bounds on the real line.",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/4ezs9y8aw6ka1",
  "venue": "HAL",
  "versions": []
}