{
  "abstract": "This article is motivated by the use of Bayesian probabilistic numerics for the construction of numerical schemes for PDEs. To avoid the \"flat limit problem\" encountered for this problem, we consider Gaussian process priors with homogeneous covariance kernels. To construct such kernels, we introduce the positive definite Fourier transform, which defines an invertible map on the set of tempered positive semi-definite distributions onto itself. This enables us to construct an extension of the fractional Brownian motion kernel with arbitrary Hurst parameter H > 0, and arbitrary space dimension R^d , d > 0. Using such kernels, we show that Gaussian process regression with such kernels and with unknown polynomial mean function is mathematically equivalent to interpolation with fractional polyharmonic splines. This finally allows us to construct numerical schemes for the scalar linear transport equation, which exhibit fractional convergence rates for non-smooth initial data.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Iain Henderson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d8qcwapskvzvz.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "d8qcwapskvzvz",
  "issns": [],
  "landing_url": "https://hal.science/hal-05749753v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-fa"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05749753/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d8qcwapskvzvz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:34Z",
      "native_id": "hal-05749753",
      "source": "hal",
      "url": "https://hal.science/hal-05749753v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-fa",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A positive-definite Fourier transform, with applications to fractional Brownian motions and Bayesian probabilistic numerics",
  "title_text": "A positive-definite Fourier transform, with applications to fractional Brownian motions and Bayesian probabilistic numerics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:34Z",
  "url": "https://research.draconitic.com/paper/d8qcwapskvzvz",
  "venue": "HAL",
  "versions": []
}