Let be the real Hilbert space equipped with the inner product
where is the rotational invariant probability measure on , and is the volume of . Denote by the usual orthonormal basis of spherical harmonics. For each fixed , the set spans the eigenspace of the Laplace-Beltrami operator on corresponding to the eigenvalue . Here is the dimension of the eigenspace corresponding to and is given by
Let , and let denote the usual dot product in .
Definition 2.1. Let . A continuous function is called a spherical basis function (SBF) of order on , if its expansion in Legendre polynomials has coefficients for all and . 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 , an SBF of order , in the following form:
with for all , and The coefficients are determined by
and will be denoted by .
Theorem 3.4. Let and let be an SBF of order on . Let be points on . Then the following three statements are equivalent:
The points are uniformly distributed on .
The following equation
holds true for each spherical harmonics with and , and the limit holds true uniformly in .
Equation (3.1) holds true for each spherical harmonics with and , and the following limit holds true
Let be an SBF of order , on . For each natural number , let denote a set on points on . We define the (normalized) point discrete -energy, , by The point discrete -energy is also realized as a function
In this paper the authors prove that minimal energy points associated with an SBF are uniformly distributed on the spheres.
Theorem 4.3. Let be an SBF of order , on , and let be a set of points on that minimizes the point discrete -energy, i.e., where the minimum is taken over the set of all possible . Then the points are uniformly distributed on .
Let be a SBF of order , , on and let be a collection of points that minimizes the -point discrete -energy. The authors use Reproducing Kernel Hilbert Space (RKHS) theory to obtain the estimate of quantity
In the last part of the paper the authors estimate the separation of the minimal energy points.