Radial basis interpolation on homogeneous manifolds: convergence rates
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Radial basis interpolation on homogeneous manifolds: convergence rates J. Levesley · D. L. Ragozin
Received: 7 August 2004 / Accepted: 4 July 2005 / Published online: 31 January 2007 © Springer Science + Business Media B.V. 2007
Abstract Pointwise error estimates for approximation on compact homogeneous manifolds using radial kernels are presented. For a C 2r positive definite kernel κ the pointwise error at x for interpolation by translates of κ goes to 0 like ρ r , where ρ is the density of the interpolating set on a fixed neighbourhood of x. Tangent space techniques are used to lift the problem from the manifold to Euclidean space, where methods for proving such error estimates are well established. Keywords scattered data interpolation · homogeneous manifold · radial basis functions · zonal kernel · convergence rates Mathematics Subject Classifications (2000) Primary 41A05 · Secondary 41A15 · 41A63 1 Introduction There is currently significant interest in approximation on spheres, related to many interesting geophysical problems. There are a number of different approximation methods currently available on spheres, including wavelets [4], piecewise polynomial splines [1], and the subject of this paper, radial functions (sometimes called zonal
Supported by EPSRC Grant GR/L36222. Partially supported by NSF Grant DMS-9972004. Communicated by J. Levesley. J. Levesley (B) Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK e-mail: [email protected] D. L. Ragozin Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195, USA e-mail: [email protected]
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J. Levesley, D.L. Ragozin
splines or variational splines) [4, 7]. Error estimates and convergence rates for radial approximation on spheres, of an optimal nature, are recent in vintage [5, 6], and rely on some technically demanding mathematics. In this paper we build on an idea of Bos and de Marchi [3] in order to provide convergence rates for radial interpolation on a much wider class of manifolds: the reflexive, compact homogeneous spaces. In fact, much of what is accomplished here can also be achieved on a Riemannian manifold; see [8]. Let M ⊂ IRd+k be a d-dimensional embedded compact homogeneous C∞ manifold in the sense of Definition 1.1 in [10]. In particular there is a compact group G of isometries of IRd+k such that for some η ∈ M (often referred to as the pole) M = {gη : g ∈ G}. The reflexive condition means that for each pair x, y ∈ M there is a g ∈ G with gx = y and gy = x. A kernel κ : M × M → IR is termed zonal (or G-invariant) if κ(x, y) = κ(gx, gy) for all g ∈ G and x, y ∈ M. Since the maps in G are isometries of Euclidean space, they preserve both Euclidean distance and the (arc-length) metric d(·, ·) induced on the components of M by the Euclidean metric. Thus the distance kernel d(x, y) is zonal, as are all the radial functions, φ(d(x, y)), which are kernels that depend only on the distance between x, y. The manifold carries a unique normalised G-invariant measure which we call μ (μ can be viewed eith
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