Constructive Approximation

, Volume 38, Issue 3, pp 397–445 | Cite as

A Characterization of Sobolev Spaces on the Sphere and an Extension of Stolarsky’s Invariance Principle to Arbitrary Smoothness



In this paper, we study reproducing kernel Hilbert spaces of arbitrary smoothness on the sphere \(\mathbb{S}^{d} \subset\mathbb{R}^{d+1}\). The reproducing kernel is given by an integral representation using the truncated power function \((\mathbf{x} \cdot\mathbf{z} - t)_{+}^{\beta-1}\) 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 ∥xy∥ 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 \(\beta\in\mathbb{N}\), 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 \(\mathbb{L}_{2}\)-discrepancy of these points remains constant regardless of the choice of the points. Rearranged differently, it provides a reinterpretation of the spherical cap \(\mathbb{L}_{2}\)-discrepancy as the worst-case error of equal-weight numerical integration rules in the Sobolev space over \(\mathbb{S}^{d}\) of smoothness (d+1)/2 provided with the reproducing kernel 1−C d xy∥ for some constant C d .

Using the new function spaces, we establish an invariance principle for a generalized discrepancy extending the spherical cap \(\mathbb{L}_{2}\)-discrepancy and give a reinterpretation as the worst-case error in the Sobolev space over \(\mathbb{S}^{d}\) 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).


Equal weight numerical integration Generalized discrepancy Reproducing kernel Hilbert space Sobolev space Sphere Spherical designs Stolarsky’s invariance principle Warnock’s formula Worst-case numerical integration error 

Mathematics Subject Classification

41A30 11K38 41A55 



The authors are grateful to Michael Gnewuch for a fruitful discussion which lead to the proof in Appendix A. Further, the authors would like to thank Yuguang Wang and two anonymous referees for their helpful comments.


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

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