{
  "id": "1908.01178",
  "version": "v1",
  "published": "2019-08-03T13:51:37.000Z",
  "updated": "2019-08-03T13:51:37.000Z",
  "title": "Function integration, reconstruction and approximation using rank-1 lattices",
  "authors": [
    "Frances Y. Kuo",
    "Giovanni Migliorati",
    "Fabio Nobile",
    "Dirk Nuyens"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We consider rank-1 lattices for integration and reconstruction of functions with series expansion supported on a finite index set. We explore the connection between the periodic Fourier space and the non-periodic cosine space and Chebyshev space, via tent transform and then cosine transform, to transfer known results from the periodic setting into new insights for the non-periodic settings. Fast discrete cosine transform can be applied for the reconstruction phase. To reduce the size of the auxiliary index set in the associated component-by-component (CBC) construction for the lattice generating vectors, we work with a bi-orthonormal set of basis functions, leading to three methods for function reconstruction in the non-periodic settings. We provide new theory and efficient algorithmic strategies for the CBC construction. We also interpret our results in the context of general function approximation and discrete least-squares approximation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-03T13:51:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A10",
      "42A10",
      "41A63",
      "42B05",
      "65D30",
      "65D32",
      "65D15"
    ],
    "keywords": [
      "function integration",
      "reconstruction",
      "non-periodic settings",
      "fast discrete cosine transform",
      "finite index set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}