Locally Supported Kernels for Spherical Spline Interpolation

Abstract

By the use of locally supported basis functions for spherical spline interpolation the applicability of this approximation method is extended, since the resulting interpolation matrix is sparse and thus efficient solvers can be used. In this paper we study locally supported kernels in detail. Investigations on the Legendre coefficients allow a characterization of the underlying Hilbert space structure. We show how spherical spline interpolation with polynomial precision can be managed with locally supported kernels, thus giving the possibility to combine approximation techniques based on spherical harmonic expansions with those based on locally supported kernels.1997 Academic Press

Multiresolution Analysis by Spherical Up Functions

“…Note that in contrast to earlier investigations of these kernels (see e.g. [28]) we let the parameter λ be real, and allow the functions to be unbounded (for −1 < λ < 0), but with finite integral. Letting η ∈ Ω be fixed, we get a radial basis function…”

“…Based on the recursion relations (3) we are able to derive recursion formulas for B ∧ h,λ (n), where we follow the lines predetermined by [22], [31], [10], [8], [28]. A straightforward integration yields…”

“…invariant pseudodifferential equations [7], regularization of inverse problems [14], etc). However, the locally supported isotropic kernel functions (in brief, isotropic finite elements) on the sphere used so far ( [10], [15], [17], [8], [28], [9], [22], [32]) possess a non-monotone L 2 (Ω)-symbol, which prevents them to be used for building up a multiresolution analysis (in the sense that the space of square-integrable functions on the sphere can be decomposed by a nested sequence of subspaces including a closure property). Furthermore, finite elements show only a limited order of differentiability on the sphere, which may be not appropriate when investigating smooth functions.…”

A Tree Algorithm for Isotropic Finite Elements on the Sphere

“…We start by repeating some formulas for L (k) h known from the investigations in e.g. [7], [20], [22], [23]. …”

Locally supported approximate identities on the unit ball

“…In [2,6] axisymmetric locally supported basis functions on the 2-sphere are introduced by use of Bernstein polynomials. Their treatment as foundation for spherical spline interpolation in certain Sobolev spaces is discussed in [14]. In [5] their application to satellite data is investigated.…”