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Abstract
We analyze explicit trial functions defined on the unit sphere S_{d} in the Euclidean space ℝ^{d+1}, d ≥ 1, that are integrable in the L_{p}sense, p ε [1,∞). These functions depend on two free parameters: one determines the support and one, a critical exponent, controls the behavior near the boundary of the support. Three noteworthy features are: (1) they are simple to implement and capture typical behavior of functions in applications, (2) their integrals with respect to the uniform measure on the sphere are given by explicit formulas and, thus, their numerical values can be computed to arbitrary precision, and (3) their smoothness can be defined a priori, that is to say, they belong to Sobolev spaces H^{s}(S^{d}) up to a specified index Ns determined by the parameters of the function.Considered are zonal functions g(x) = h(x · p), where p is some fixed pole on S^{d}. The function h(t) is of the type [max(t, T)]^{α} or a variation of a truncated power function x (mapping) (x)^{α} _{+} C (which assumes 0 if x ≤ 0 and is the power x^{α} if x > 0) that reduces to [max(t  T,0)]α, [max(t^{2}  T^{2}, 0)]^{α}, and [max(T^{2}  t_{2}, 0)]^{α} if α > 0. These types of trial functions have as support the whole sphere, a spherical cap centered at p, a bicap centered at the antipodes p, p, or an equatorial belt. We give inclusion theorems that identify the critical smoothness s = s(T, α) and explicit formulas for the integral over the sphere. We obtain explicit formulas for the coefficients in the Laplace Fourier expansion of these trial functions and provide the leading order term in the asymptotics for large index of the coefficients.
Original language  English 

Title of host publication  Contemporary Computational Mathematics  A Celebration of the 80th Birthday of Ian Sloan 
Publisher  Springer International Publishing AG 
Pages  153177 
Number of pages  25 
ISBN (Electronic)  9783319724560 
ISBN (Print)  9783319724553 
DOIs  
Publication status  Published  23 May 2018 
ASJC Scopus subject areas
 Mathematics(all)
Fields of Expertise
 Information, Communication & Computing
Treatment code (Nähere Zuordnung)
 Basic  Fundamental (Grundlagenforschung)
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Projects
 1 Finished

FWF  Self WW  Self organization by local interaction: minimal energy, external fields, and numerical integration
1/10/16 → 28/02/19
Project: Research project