{
  "id": "1304.0328",
  "version": "v1",
  "published": "2013-04-01T10:43:24.000Z",
  "updated": "2013-04-01T10:43:24.000Z",
  "title": "Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order",
  "authors": [
    "Josef Dick"
  ],
  "journal": "J. Dick, Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal., 46, 1519--1553, 2008",
  "doi": "10.1137/060666639",
  "categories": [
    "math.NA"
  ],
  "abstract": "We define a Walsh space which contains all functions whose partial mixed derivatives up to order $\\delta \\ge 1$ exist and have finite variation. In particular, for a suitable choice of parameters, this implies that certain Sobolev spaces are contained in these Walsh spaces. For this Walsh space we then show that quasi-Monte Carlo rules based on digital $(t,\\alpha,s)$-sequences achieve the optimal rate of convergence of the worst-case error for numerical integration. This rate of convergence is also optimal for the subspace of smooth functions. Explicit constructions of digital $(t,\\alpha,s)$-sequences are given hence providing explicit quasi-Monte Carlo rules which achieve the optimal rate of convergence of the integration error for arbitrarily smooth functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-04-01T10:43:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K38",
      "11K45",
      "65C05",
      "42C10"
    ],
    "keywords": [
      "walsh spaces containing smooth functions",
      "quasi-monte carlo rules",
      "arbitrary high order"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.0328D"
    }
  }
}