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

Abstract

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 $(\boldsymbol{x} \cdot \boldsymbol{z} - t)_+^{\beta-1}$ defined on spherical caps centered at $\boldsymbol{z}$ of height $t$, which reduce to an integral over indicator functions of spherical caps as studied in [J. Brauchart, J. Dick, arXiv:1101.4448v1 [math.NA], to appear in Proc. Amer. Math. Soc.] for $\beta = 1$. This is in analogy to the generalization of the reproducing kernel to arbitrary smoothness on the unit cube. We show that the reproducing kernel is a sum of a Kamp{\'e} de F{\'e}riet function and the Euclidean distance $\|\boldsymbol{x}-\boldsymbol{y}\|$ of the arguments of the kernel raised to the power of $2\beta -1$ if $2\beta - 1$ is not an even integer; otherwise the logarithm of the distance $\|\boldsymbol{x}-\boldsymbol{y}\|$ appears. For $\beta \in \mathbb{N}$ the Kamp\'e de F\'eriet function reduces to a polynomial, giving a simple closed form expression for the reproducing kernel. Using this space we can generalize Stolarsky's invariance principle to arbitrary smoothness. Previously, Warnock's formula, which is the analogue to Stolarsky's invariance principle for the unit cube $[0,1]^s$, has been generalized using similar techniques [J. Dick, Ann. Mat. Pura. Appl., (4) 187 (2008), no. 3, 385--403].