# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Spherical basis functions and uniform distribution of points on spheres. (English) Zbl 1148.41028

Let ${L}^{2}\left({S}^{d}\right)$ be the real Hilbert space equipped with the inner product

$〈f,g〉={\omega }_{d}{\int }_{{S}^{d}}f\left(x\right)g\left(x\right)\phantom{\rule{0.166667em}{0ex}}d\sigma \left(x\right),$

where $d\sigma$ is the rotational invariant probability measure on ${S}^{d}$, and ${\omega }_{d}$ is the volume of ${S}^{d}$. Denote by ${Y}_{l,m}$ the usual orthonormal basis of spherical harmonics. For each fixed $l$, the set $\left\{{Y}_{l,m}:m=1,\cdots ,{q}_{l}\right\}$ spans the eigenspace of the Laplace-Beltrami operator on ${S}^{d}$ corresponding to the eigenvalue ${\lambda }_{l}=l\left(l+d-1\right)$. Here ${q}_{l}$ is the dimension of the eigenspace corresponding to ${\lambda }_{l}$ and is given by

${q}_{l}=\frac{\left(2l+d-1\right){\Gamma }\left(l+d-1\right)}{{\Gamma }\left(l+1\right){\Gamma }\left(d\right)},\phantom{\rule{1.em}{0ex}}l\ge 1·$

Let $x,y\in {S}^{d}$, and let $x·y$ denote the usual dot product in ${ℝ}^{d+1}$.

Definition 2.1. Let $k\ge 0$. A continuous function $\phi :\left[-1,1\right]\to ℝ$ is called a spherical basis function (SBF) of order $k$ on ${S}^{d}$, if its expansion in Legendre polynomials $\phi \left(x·y\right)={\sum }_{l=0}^{\infty }{a}_{l}\frac{{q}_{l}}{{\omega }_{d}}{P}_{l}^{\left(\nu \right)}\left(x·y\right),$ has coefficients ${a}_{l}>0$ for all $l\ge k$ and $\sum {a}_{l}{q}_{l}<\infty$. An SBF of order 0 will simply be called an SBF.

In this paper the authors characterize uniform distribution of points on spheres in terms of SBFs. The authors use the summation formula for spherical harmonics to write $\phi$, an SBF of order $k$, in the following form:

$\phi \left(x·y\right)=\sum _{l=0}^{\infty }{a}_{l}\sum _{m=1}^{{q}_{l}}{Y}_{l,m}\left(x\right){Y}_{l,m}\left(y\right)$

with ${a}_{l}>0$ for all $l\ge k$, and $\sum {a}_{l}{q}_{l}<\infty ·$ The coefficients ${a}_{l}$ are determined by

${a}_{l}={\omega }_{d}^{2}{\int }_{{S}^{d}}{\int }_{{S}^{d}}\phi \left(x·y\right){Y}_{l,m}\left(x\right){Y}_{l,m}\left(y\right)\phantom{\rule{0.166667em}{0ex}}d\sigma \left(x\right)\phantom{\rule{0.166667em}{0ex}}d\sigma \left(y\right),$

and ${a}_{0}$ will be denoted by ${A}_{\phi }$.

Theorem 3.4. Let $k\ge 0$ and let $\phi$ be an SBF of order $k$ on ${S}^{d}$. Let ${x}_{1},\cdots ,{x}_{N}$ be $N$ points on ${S}^{d}$. Then the following three statements are equivalent:

The points ${x}_{1},\cdots ,{x}_{N}$ are uniformly distributed on ${S}^{d}$.

The following equation

$\underset{N\to \infty }{lim}\frac{1}{N}\sum _{j=1}^{N}{Y}_{l,m}\left({x}_{j}\right)=0\phantom{\rule{2.em}{0ex}}\left(3·1\right)$

holds true for each spherical harmonics ${Y}_{l,m}$ with $1\le l and $m=1,\cdots ,{q}_{l}$, and the limit ${lim}_{N\to \infty }\frac{1}{N}{\sum }_{j=1}^{N}\phi \left({x}_{j}·y\right)={A}_{\phi }$ holds true uniformly in $y\in {S}^{d}$.

Equation (3.1) holds true for each spherical harmonics ${Y}_{l,m}$ with $1\le l and $m=1,\cdots ,{q}_{l}$, and the following limit holds true

$\underset{N\to \infty }{lim}\frac{1}{{N}^{2}}\sum _{i=1}^{N}\sum _{j=1}^{N}\phi \left({x}_{i}·{x}_{j}\right)={A}_{\phi }·$

Let $\phi$ be an SBF of order $k$, $k=0,1,$ on ${S}^{d}$. For each natural number $N$, let ${{\Omega }}_{N}={\left\{{x}_{j}\right\}}_{j=1}^{N}$ denote a set on $N$ points on ${S}^{d}$. We define the (normalized) $N$ point discrete $\phi$-energy, ${E}_{\phi }\left({{\Omega }}_{N}\right)$, by ${E}_{\phi }\left({{\Omega }}_{N}\right)=\frac{1}{{N}^{2}}{\sum }_{i=1}^{N}{\sum }_{j=1}^{N}\phi \left({x}_{i}·{x}_{j}\right)·$ The $N$ point discrete $\phi$-energy is also realized as a function

${E}_{\phi }\left({{\Omega }}_{N}\right):\underset{N}{\underbrace{{S}^{d}×\cdots ×{S}^{d}}}\to ℝ·$

In this paper the authors prove that minimal energy points associated with an SBF are uniformly distributed on the spheres.

Theorem 4.3. Let $\phi$ be an SBF of order $k$, $k=0,1$ on ${S}^{d}$, and let ${{\Omega }}_{N}^{*}=\left\{{x}_{1}^{*},\cdots ,{x}_{N}^{*}\right\}$ be a set of $N$ points on ${S}^{d}$ that minimizes the $N$ point discrete $\phi$-energy, i.e., ${E}_{\phi }\left({{\Omega }}_{N}^{*}\right)={min}_{{{\Omega }}_{N}}{E}_{\phi }\left({{\Omega }}_{N}\right),$ where the minimum is taken over the set of all possible ${{\Omega }}_{N}$. Then the points ${x}_{1}^{*},\cdots ,{x}_{N}^{*}$ are uniformly distributed on ${S}^{d}$.

Let $\phi$ be a SBF of order $k$, $k=0,1$, on ${S}^{d}$ and let ${x}_{1},\cdots ,{x}_{N}$ be a collection of $N$ points that minimizes the $N$-point discrete $\phi$-energy. The authors use Reproducing Kernel Hilbert Space (RKHS) theory to obtain the estimate of quantity

$\left|\frac{1}{N}\sum _{j=1}^{N}f\left({x}_{j}\right)-{\int }_{{S}^{d}}f\left(x\right)\phantom{\rule{0.166667em}{0ex}}d\sigma \left(x\right)\right|·$

In the last part of the paper the authors estimate the separation of the minimal energy points.

##### MSC:
 41A55 Approximate quadratures 11K38 Irregularities of distribution 11K41 Continuous, $p$-adic and abstract analogues 46E22 Hilbert spaces with reproducing kernels