{
  "id": "1104.1709",
  "version": "v1",
  "published": "2011-04-09T14:13:55.000Z",
  "updated": "2011-04-09T14:13:55.000Z",
  "title": "Variational splines on Riemannian manifolds with applications to integral geometry",
  "authors": [
    "Isaac Pesenson"
  ],
  "comment": "Adv. in Appl. Math. 33 (2004), no. 3, 548-572",
  "categories": [
    "math.FA"
  ],
  "abstract": "We extend the classical theory of variational interpolating splines to the case of compact Riemannian manifolds. Our consideration includes in particular such problems as interpolation of a function by its values on a discrete set of points and interpolation by values of integrals over a family of submanifolds. The existence and uniqueness of interpolating variational spline on a Riemannian manifold is proven. Optimal properties of such splines are shown. The explicit formulas of variational splines in terms of the eigen functions of Laplace-Beltrami operator are found. It is also shown that in the case of interpolation on discrete sets of points variational splines converge to a function in $C^{k}$ norms on manifolds. Applications of these results to the hemispherical and Radon transforms on the unit sphere are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-04-09T14:13:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "integral geometry",
      "applications",
      "discrete set",
      "points variational splines converge",
      "compact riemannian manifolds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1104.1709P"
    }
  }
}