A characterization of Sobolev spaces on the sphere and an extension of Stolarsky's invariance principle to arbitrary smoothness

Johann Brauchart*, Josef Dick

*Korrespondierende/r Autor/-in für diese Arbeit

Publikation: Beitrag in einer FachzeitschriftArtikelBegutachtung

Abstract

In this paper, we study reproducing kernel Hilbert spaces of arbitrary
smoothness on the sphere Sd ⊂ Rd+1 . The reproducing kernel is given by an integral representation using the truncated power function (x · z − t)
β−1 + supported on spherical caps centered at z of height t, which reduces to an integral over indicator functions of open spherical caps if β = 1, as studied in Brauchart and Dick (Proc. Am. Math. Soc. 141(6):2085–2096, 2013). This is analogous to a generalization of the reproducing kernel to arbitrary smoothness on the unit cube by Temlyakov (J. Complex. 19(3):352–391, 2003).
We show that the reproducing kernel is a sum of the Euclidean distance ‖x − y‖
of the arguments of the kernel raised to the power of 2β − 1 and an adjustment in
the form of a Kampé de Fériet function that ensures positivity of the kernel if 2β − 1 is not an even integer; otherwise, a limit process introduces logarithmic terms in the distance. For β ∈ N, the Kampé de Fériet function reduces to a polynomial, giving a simple closed form expression for the reproducing kernel.
Stolarsky’s invariance principle states that the sum of all mutual distances among
N points plus a certain multiple of the spherical cap L2 -discrepancy of these points remains constant regardless of the choice of the points. Rearranged differently, it provides a reinterpretation of the spherical cap L2 -discrepancy as the worst-case error of equal-weight numerical integration rules in the Sobolev space over Sd of smoothness (d + 1)/2 provided with the reproducing kernel 1 − Cd ‖x − y‖ for some constant C d .
Using the new function spaces, we establish an invariance principle for a gener-
alized discrepancy extending the spherical cap L2 -discrepancy and give a reinterpretation as the worst-case error in the Sobolev space over Sd of arbitrary smoothness s = β − 1/2 + d/2. Previously, Warnock’s formula, which is the analog to Stolarsky’s invariance principle for the unit cube [0, 1]s , has been generalized using similar techniques in Dick (Ann. Mat. Pura Appl. (4) 187(3):385–403, 2008).
Originalspracheenglisch
Seiten (von - bis)397-445
FachzeitschriftConstructive Approximation
Jahrgang38
Ausgabenummer3
DOIs
PublikationsstatusVeröffentlicht - 2013

Fields of Expertise

  • Information, Communication & Computing

Treatment code (Nähere Zuordnung)

  • Basic - Fundamental (Grundlagenforschung)
  • Application
  • Theoretical

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