{
  "id": "2206.07432",
  "version": "v1",
  "published": "2022-06-15T10:04:50.000Z",
  "updated": "2022-06-15T10:04:50.000Z",
  "title": "A short note on compact embeddings of reproducing kernel Hilbert spaces in $L^2$ for infinite-variate function approximation",
  "authors": [
    "Marcin Wnuk"
  ],
  "categories": [
    "math.FA",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "This note consists of two largely independent parts. In the first part we give conditions on the kernel $k: \\Omega \\times \\Omega \\rightarrow \\mathbb{R}$ of a reproducing kernel Hilbert space $H$ continuously embedded via the identity mapping into $L^2(\\Omega, \\mu),$ which are equivalent to the fact that $H$ is even compactly embedded into $L^2(\\Omega, \\mu).$ In the second part we consider a scenario from infinite-variate $L^2$-approximation. Suppose that the embedding of a reproducing kernel Hilbert space of univariate functions with reproducing kernel $1+k$ into $L^2(\\Omega, \\mu)$ is compact. We provide a simple criterion for checking compactness of the embedding of a reproducing kernel Hilbert space with the kernel given by $$\\sum_{u \\in \\mathcal{U}} \\gamma_u \\bigotimes_{j \\in u}k,$$ where $\\mathcal{U} = \\{u \\subset \\mathbb{N}: |u| < \\infty\\},$ and $(\\gamma_u)_{u \\in \\mathcal{U}}$ is a sequence of non-negative numbers, into an appropriate $L^2$ space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-15T10:04:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "reproducing kernel hilbert space",
      "infinite-variate function approximation",
      "compact embeddings",
      "short note",
      "largely independent parts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}