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

Mathematics – Numerical Analysis

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

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].

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

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

     

Profile ID: LFWR-SCP-O-471502

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.