Energy functionals, numerical integration and asymptotic equidistribution on the sphere.
The authors study equally weighted quadrature rules for the numerical integration of continuous functions on \(d\)-spheres \(S^d\). The quadrature knots are choosen to minimize some energy functional \[ E(g, Z_N)= {1\over N} \sum^N_{\substack{ i,j=1\\ i\neq j}} g(\langle x_i, x_j\rangle) \] with respect to the associated point set \(Z_N\). A point set for which the minimal energy is attained, is called a \(g\)-minimal enery point set. Here the function \(g\) has to fulfill some admissibility conditions and typical admissible functions are \[ g^0_L(t)= {1\over 2}\log{1\over 1-t}- {1\over 2}\log 2 \] and \[ g^s_R(t)= {1\over 2^{s/2}(1-t)^{s/2}} \quad (s> 0) \] for the logarithmic energy and the energy corresponding to the Riesz potential \({1\over r^s}\), respectively. The authors prove some quadrature error estimates involving \(E(g, Z_N)\) for arbitrary point distributions \(Z_N\). For \(0\leq s< d\), they conclude that the \(g^s_R\)-minimal energy point sets are asymptotically equidistributed as \(N\to\infty\) and mention that this result is well known. For \(s= d\) they present explicit rates of convergence which had been open up to now and prove that \(g^d_R\)-minimal energy points are asymptotically equidistributed with rate \(1/\sqrt{\log N}\). [An estimate in theorem 4 is corrected.]
scientific article; zbMATH DE number 1967205
scientific article