A simple Proof of Stolarsky's Invariance Principle

Stolarsky [Proc. Amer. Math. Soc. 41 (1973), 575--582] showed a beautiful relation that balances the sums of distances of points on the unit sphere and their spherical cap $\mathbb{L}_2$-discrepancy to give the distance integral of the uniform measure on the sphere a potential-theoretical quantity (Bj{\"o}rck [Ark. Mat. 3 (1956), 255--269]). Read differently it expresses the worst-case numerical integration error for functions from the unit ball in a certain Hilbert space setting in terms of the $\mathbb{L}_2$-discrepancy and vice versa (first author and Womersley [Preprint]). In this note we give a simple proof of the invariance principle using reproducing kernel Hilbert spaces

An Electrostatics Problem on the Sphere Arising from a Nearby Point Charge

For a positively charged insulated d-dimensional sphere we investigate how the distribution of this charge is affected by proximity to a nearby positive or negative point charge when the system is governed by a Riesz s-potential 1/r^s, s>0, where r denotes Euclidean distance between point charges. Of particular interest are those distances from the point charge to the sphere for which the equilibrium charge distribution is no longer supported on the whole of the sphere (i.e. spherical caps of negative charge appear). Arising from this problem attributed to A. A. Gonchar are sequences of polynomials of a complex variable that have some fascinating properties regarding their zeros.Comment: 44 pages, 9 figure

Riesz external field problems on the hypersphere and optimal point separation

We consider the minimal energy problem on the unit sphere $\mathbb{S}^d$ in the Euclidean space $\mathbb{R}^{d+1}$ in the presence of an external field $Q$, where the energy arises from the Riesz potential $1/r^s$ (where $r$ is the Euclidean distance and $s$ is the Riesz parameter) or the logarithmic potential $\log(1/r)$. Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range $d-2 \leq s < d - 1.$ The proof uses a maximum principle for measures supported on $\mathbb{S}^d$. When $Q$ is the Riesz $s$-potential of a signed measure and $d-2 \leq s <d$, our results lead to explicit point-separation estimates for $(Q,s)$-Fekete points, which are $n$-point configurations minimizing the Riesz $s$-energy on $\mathbb{S}^d$ with external field $Q$. In the hyper-singular case $s > d$, the short-range pair-interaction enforces well-separation even in the presence of more general external fields. As a further application, we determine the extremal and signed equilibria when the external field is due to a negative point charge outside a positively charged isolated sphere. Moreover, we provide a rigorous analysis of the three point external field problem and numerical results for the four point problem.Comment: 35 pages, 4 figure

Quasi-Monte Carlo rules for numerical integration over the unit sphere S2\mathbb{S}^2

We study numerical integration on the unit sphere $\mathbb{S}^2 \subset \mathbb{R}^3$ using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly. The quadrature points are constructed by lifting a $(0,m,2)$-net given in the unit square $[0,1]^2$ to the sphere $\mathbb{S}^2$ by means of an area preserving map. A similar approach has previously been suggested by Cui and Freeden [SIAM J. Sci. Comput. 18 (1997), no. 2]. We prove three results. The first one is that the construction is (almost) optimal with respect to discrepancies based on spherical rectangles. Further we prove that the point set is asymptotically uniformly distributed on $\mathbb{S}^2$. And finally, we prove an upper bound on the spherical cap $L_2$-discrepancy of order $N^{-1/2} (\log N)^{1/2}$ (where $N$ denotes the number of points). This slightly improves upon the bound on the spherical cap $L_2$-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Comm. Pure Appl. Math. 39 (1986), 149--186]. Numerical results suggest that the $(0,m,2)$-nets lifted to the sphere $\mathbb{S}^2$ have spherical cap $L_2$-discrepancy converging with the optimal order of $N^{-3/4}$

Numerical integration over spheres of arbitrary dimension

In this paper, we study the worst-case error (of numerical integration) on the unit sphere $\\mathbb{S}^{d}$, $d\\geq 2$, for all functions in the unit ball of the Sobolev space $\\mathbb{H}^s(\\mathbb{S}^d)$, where s>d/2. More precisely, we consider infinite sequences $(Q_{m(n)})_{n\\in\\mathbb{N}}$ of $m(n)$-point numerical integration rules $Q_{m(n)}$, where (i) $Q_{m(n)}$ is exact for all spherical polynomials of degree $\\leq n$, and (ii) $Q_{m(n)}$ has positive weights or, alternatively to (ii), the sequence $(Q_{m(n)})_{n\\in\\mathbb{N}}$ satisfies a certain local regularity property. Then we show that the worst-case error (of numerical integration) $E(Q_{m(n)};\\mathbb{H}^s(\\matbb{S}^d))$ in $\\mathbb{H}^s(\\mathbb{S}^d)$ has the upper bound $c n^{-s}$, where the constant $c$ depends on $s$ and $d$ (and possibly the sequence $(Q_{m(n)})_{n\\in\\mathbb{N}}$). This extends the recent results for the sphere $\\mathbb{S}^2$ by K.Hesse and I.H.Sloan to spheres $\\mathbb{S}^d$ of arbitrary dimension $d\\geq2$ by using an alternative representation of the worst-case error. If the sequence $(Q_{m(n)})_{n\\in\\mathbb{N}}$ of numerical integration rules satisfies $m(n)=\\mathcal{O}(n^d)$ an order-optimal rate of convergence is achieved