Localized polynomial bases on the sphere (Q2388697)

Let \(V_n\) be the space of polynomials of degree not greater than \(n\) on the unit sphere \(S^2\) of \(R^3\). The dimension of \(V_n\) is \(N=(n+1)^2\) and its reproducing kernel is \[ K_n(\xi,\eta)=\sum_{l=0}^n {{2l+1}\over {4\pi}}P_l(\xi\cdot \eta). \] Here \(P_l\) denotes the Legendre polynomial of degree \(l\) normalized by \(P_l(1)=1\) and \(\xi\cdot \eta\) the Euclidean product. The aim of this paper is the characterization and construction of systems of \(N\) points \(\{\xi_i\}_{i=1}^N \subset S^2\) such that the scaling functions \(\{\phi_i^n:=K_n(\xi_i,\cdot)\}_{i=1}^N\) form a basis of \(V_n\).