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Spherical basis functions and uniform distribution of points on spheres. (English) Zbl 1148.41028

Let \(L^{2}(S^{d})\) be the real Hilbert space equipped with the inner product
\[ \langle f,g \rangle=\omega_{d} \int_{S^{d}} f(x)g(x) \,d\sigma(x), \] 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 \(\{ Y_{l,m}: m=1,\dots, q_{l}\}\) spans the eigenspace of the Laplace-Beltrami operator on \(S^{d}\) corresponding to the eigenvalue \(\lambda_{l}=l(l+d-1)\). Here \(q_{l}\) is the dimension of the eigenspace corresponding to \(\lambda_{l}\) and is given by
\[ q_{l}= \frac{(2l+d-1)\Gamma(l+d-1)}{\Gamma(l+1)\Gamma(d)}, \quad l\geq 1. \]
Let \(x,y\in S^{d}\), and let \(x\cdot y\) denote the usual dot product in \(\mathbb R^{d+1}\).
Definition 2.1. Let \(k\geq 0\). A continuous function \(\varphi:[-1,1]\to\mathbb R\) is called a spherical basis function (SBF) of order \(k\) on \(S^{d}\), if its expansion in Legendre polynomials \(\varphi(x\cdot y)=\sum_{l=0}^{\infty} a_{l} \frac{q_{l}}{\omega_{d}}P_{l}^{(\nu)}(x\cdot y),\) has coefficients \(a_{l}>0\) for all \(l\geq k\) 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 \(\varphi\), an SBF of order \(k\), in the following form:
\[ \varphi(x\cdot y)= \sum_{l=0}^{\infty}a_{l} \sum_{m=1}^{q_{l}}Y_{l,m}(x)Y_{l,m}(y) \]
with \(a_{l}>0\) for all \(l\geq k\), and \(\sum a_{l}q_{l}<\infty.\) The coefficients \(a_{l}\) are determined by
\[ a_l=\omega_d^2 \int_{S^d} \int_{S^d} \varphi(x\cdot y) Y_{l,m}(x) Y_{l,m}(y)\,d\sigma(x)\,d\sigma(y), \]
and \(a_{0}\) will be denoted by \(A_\varphi\).
Theorem 3.4. Let \(k\geq 0\) and let \(\varphi\) be an SBF of order \(k\) on \(S^{d}\). Let \(x_{1},\dots,x_{N}\) be \(N\) points on \(S^{d}\). Then the following three statements are equivalent:
1.
The points \(x_{1},\dots,x_{N}\) are uniformly distributed on \(S^{d}\).
2.
The following equation
\[ \lim_{N\to \infty} \frac{1}{N}\sum_{j=1}^{N}Y_{l,m}(x_{j})=0\tag{3.1} \]
holds true for each spherical harmonics \(Y_{l,m}\) with \(1\leq l<k\) and \(m=1,\dots, q_{l}\), and the limit \(\lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^{N}\varphi(x_{j}\cdot y)=A_{\varphi}\) holds true uniformly in \(y\in S^{d}\).
3.
Equation ({3.1}) holds true for each spherical harmonics \(Y_{l,m}\) with \(1\leq l< k\) and \(m=1,\dots, q_{l}\), and the following limit holds true
\[ \lim_{N\to\infty}\frac{1}{N^{2}} \sum_{i=1}^{N} \sum_{j=1}^{N} \varphi(x_{i}\cdot x_{j})=A_\varphi. \]
Let \(\varphi\) be an SBF of order \(k\), \(k=0,1,\) on \(S^{d}\). For each natural number \(N\), let \( \Omega_{N}=\{ x_{j}\}_{j=1}^{N}\) denote a set on \(N\) points on \(S^{d}\). We define the (normalized) \(N\) point discrete \(\varphi\)-energy, \(E_\varphi(\Omega_{N})\), by \(E_\varphi(\Omega_{N})= \frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{j=1}^{N}\varphi(x_{i}\cdot x_{j}).\) The \(N\) point discrete \(\varphi\)-energy is also realized as a function
\[ E_\varphi(\Omega_{N}): \underbrace{S^{d}\times\cdots\times S^{d}}_{N}\to\mathbb R. \]
In this paper the authors prove that minimal energy points associated with an SBF are uniformly distributed on the spheres.
Theorem 4.3. Let \(\varphi\) be an SBF of order \(k\), \(k=0,1\) on \(S^{d}\), and let \(\Omega_{N}^{*}=\{x_{1}^{*},\dots, x_{N}^{*}\}\) be a set of \(N\) points on \(S^{d}\) that minimizes the \(N\) point discrete \(\varphi\)-energy, i.e., \(E_\varphi(\Omega_{N}^{*})= \min_{\Omega_{N}}E_\varphi(\Omega_{N}),\) where the minimum is taken over the set of all possible \(\Omega_{N}\). Then the points \(x_{1}^{*},\dots,x_{N}^{*}\) are uniformly distributed on \(S^{d}\).
Let \(\varphi\) be a SBF of order \(k\), \(k=0,1\), on \(S^{d}\) and let \(x_{1},\dots,x_{N}\) be a collection of \(N\) points that minimizes the \(N\)-point discrete \(\varphi\)-energy. The authors use Reproducing Kernel Hilbert Space (RKHS) theory to obtain the estimate of quantity
\[ \left|\frac{1}{N}\sum_{j=1}^{N}f(x_{j})- \int_{S^{d}} f(x)\,d\sigma(x)\right| . \]
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, discrepancy
11K41 Continuous, \(p\)-adic and abstract analogues
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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