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Explicit Families of Functions on the Sphere with Exactly Known Sobolev Space Smoothness

  • Johann S. Brauchart
Chapter

Abstract

We analyze explicit trial functions defined on the unit sphere \({\mathbb {S}}^d\) in the Euclidean space \({\mathbb {R}}^{d+1}\), d ≥ 1, that are integrable in the \({\mathbb {L}}_p\)-sense, p ∈ [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 \({\mathbb {H}}^s({\mathbb {S}}^d)\) up to a specified index \(\bar {s}\) determined by the parameters of the function. Considered are zonal functions g(x) = h(x ⋅p), where p is some fixed pole on \({\mathbb {S}}^d\). The function h(t) is of the type \([ \max \{t,T\} ]^\alpha \) or a variation of a truncated power function \(x \mapsto (x)_+^\alpha \) (which assumes 0 if x ≤ 0 and is the power xα if x > 0) that reduces to \([ \max \{t-T,0\} ]^\alpha \), \([ \max \{t^2-T^2,0\} ]^{\alpha }\), and \([ \max \{T^2-t^2,0\} ]^{\alpha }\) if α > 0. These types of trial functions have as support the whole sphere, a spherical cap centered at p, a bi-cap centered at the antipodes p, −p, or an equatorial belt. We give inclusion theorems that identify the critical smoothness \(\bar {s} = \bar {s}(T,\alpha )\) 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.

Notes

Acknowledgements

The research of this author was supported, in part, by the Austrian Science Fund FWF project F5510 (part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications”) and M2030 Lise Meitner Programm “Self organization by local interaction”.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Analysis and Number TheoryGraz University of TechnologyGrazAustria

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