{
  "id": "1111.1013",
  "version": "v1",
  "published": "2011-11-03T23:42:55.000Z",
  "updated": "2011-11-03T23:42:55.000Z",
  "title": "Better bases for kernel spaces",
  "authors": [
    "E. J. Fuselier",
    "T. C. Hangelbroek",
    "F. J. Narcowich",
    "J. D. Ward",
    "G. B. Wright"
  ],
  "comment": "26 pages, 5 figures, 3 tables",
  "categories": [
    "math.NA",
    "math.CA"
  ],
  "abstract": "In this article we investigate the feasibility of constructing stable, local bases for computing with kernels. In particular, we are interested in constructing families $(b_{\\xi})_{\\xi\\in\\Xi}$ that function as bases for kernel spaces $S(k,\\Xi)$ so that each basis function is constructed using very few kernels. In other words, each function $b_{\\zeta}(x) = \\sum_{\\xi\\in\\Xi} A_{\\zeta,\\xi} k(x,\\xi)$ is a linear combination of samples of the kernel with few nonzero coefficients $A_{\\zeta,\\xi}$. This is reminiscent of the construction of the B-spline basis from the family of truncated power functions. We demonstrate that for a large class of kernels (the Sobolev kernels as well as many kernels of polyharmonic and related type) such bases exist. In fact, the basis elements can be constructed using a combination of roughly $O(\\log N)^d$ kernels, where $d$ is the local dimension of the manifold and $N$ is the dimension of the kernel space (i.e. $N=#\\Xi$). Viewing this as a preprocessing step -- the construction of the basis has computational cost $O(N(\\log N)^d)$. Furthermore, we prove that the new basis is $L_p$ stable and satisfies polynomial decay estimates that are stationary with respect to the density of $\\Xi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-03T23:42:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65D05",
      "65F08",
      "41A05"
    ],
    "keywords": [
      "kernel space",
      "better bases",
      "satisfies polynomial decay estimates",
      "construction",
      "b-spline basis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.1013F"
    }
  }
}