{
  "abstract": "In this work, we investigate the validity of the Szegö type strong asymptotic formula for the Sobolev orthogonal polynomials associated with a Sobolev-type inner product depending on a parameter $s$, which is defined as \\begin{equation*} \\langle f,g\\rangle _{\\fracλ{n^s}}:= \\int_{\\mathbb{T}} f(z)\\overline{g(z)} dμ_1(z)+ \\dfracλ{n^s} \\int_{\\mathbb{T}} f^{\\prime}(z)\\overline{g^{\\prime}(z)} dμ_2(z), \\quad \\text{where $s$ is fixed and } \\quad s\\geq 2, \\end{equation*} and $λ>0$. Here, $μ_1$, $μ_2$ are absolutely continuous with respect to the normalized lebesgue measure and $\\int_{\\mathbb{T}} μ^{\\prime}_i(z) dm=1$, where $μ_i^{\\prime}(z)= \\dfrac{dμ_i}{dm} >0$ for $i=1,2$. Further, $μ_1, μ_2$ belongs to the Szegö class and also has non-zero bounded Radon-Nikodym derivatives on the unit circle. %satisfies $ 0 <m \\leq \\dfrac{μ^{\\prime}_1(e^{ιθ})}{μ^{\\prime}_2(e^{ιθ})}\\leq M < \\infty $. We observe that the measure appearing in the Szegö function of the strong asymptotic limit depends on $s$. As a specific case, we consider the Bernstein-Christoffel Sobolev orthogonal polynomials and study their properties, including the asymptotic distribution and location of their zeros for different values of $s$. We further establish a spectral characterization of their zeros as the eigenvalue of an explicitly constructed modification of the associated Hessenberg matrix and provide numerical evidence supporting the obtained asymptotic and zeros results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31994",
  "author_total": 2,
  "authors": [
    {
      "name": "Sourav Majumder"
    },
    {
      "name": "A. Swaminathan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cz2wpbeywsck0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31994",
  "fulltext_url": null,
  "id": "cz2wpbeywsck0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31994",
  "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.CA"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7223705690",
  "pdf_url": "https://arxiv.org/pdf/2610.31994",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cz2wpbeywsck0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.31994",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31994"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sobolev Orthogonal Polynomials on the unit circle with varying inner product",
  "title_text": "Sobolev Orthogonal Polynomials on the unit circle with varying inner product",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/cz2wpbeywsck0",
  "venue": "arXiv",
  "versions": []
}