{
  "id": "2208.14779",
  "version": "v1",
  "published": "2022-08-31T11:34:44.000Z",
  "updated": "2022-08-31T11:34:44.000Z",
  "title": "Necessary and sufficient conditions for a family of continuous functions to form a Karhunen-Loève basis",
  "authors": [
    "Ricardo Carrizo Vergara"
  ],
  "comment": "3 pages short result",
  "categories": [
    "math.FA",
    "math.PR"
  ],
  "abstract": "Given an orthonormal system of $L^{2}(D)$ consistent of continuous functions $(f_{n})_{n}$, with $D \\subset \\mathbb{R}^{d}$ compact, and given a sequence of strictly positive coefficients $(\\lambda_{n})_{n}$ forming a convergent series, we prove that they consist in the eigenfunctions and eigenvectors of a covariance operator associated to a continuous positive-definite Kernel if and only if the sequence of partial sums $ \\sum_{j \\leq n} \\lambda_{j} f_{j}^{2} $ is equicontinuous over $D$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-31T11:34:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuous functions",
      "karhunen-loève basis",
      "sufficient conditions",
      "orthonormal system",
      "convergent series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 3,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}