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

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

2012-03-23

arxiv.org/abs/1203.5157v1

Mathematics

Numerical Analysis

Scientific paper

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].

Affiliated with

Also associated with

No associations

Rating

A characterization of Sobolev spaces on the sphere and an extension of Stolarsky's invariance principle to arbitrary smoothness does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A characterization of Sobolev spaces on the sphere and an extension of Stolarsky's invariance principle to arbitrary smoothness, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A characterization of Sobolev spaces on the sphere and an extension of Stolarsky's invariance principle to arbitrary smoothness will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-471502