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 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).