Approximating Probability Measures on Manifolds via Radial Basis Functions

Summary

Approximating a given probability measure by a sequence of normal- ized counting measures is an interesting problem and has broad applications in many areas of mathematics and engineering. If the target measure is the uniform distri- bution on a manifold then such approximation gives rise to the theory of uniform distribution of point sets and the corresponding discrepancy estimates. If the target measure is the equilibrium measure on a manifold, then such approximation leads to the minimization of certain energy functionals, which have applications in dis- cretization of manifolds, best possible site selection for polynomial interpolation and Monte Carlo method, among others. Traditionally, polynomials are the major tool in this arena, as have been demonstrated in the celebrated Weyl’s criterion, Erd?os- Turán inequalities. Recently, the novel approach of employing radial basis functions (RBFs) has been successful, especially in higher dimensional manifolds. In its gen- eral methodology, RBFs provide an efficient vehicle that allows a certain type of linear translation operators to act in various function spaces, including reproducing kernel Hilbert spaces (RKHS) associated with RBFs. This approach is crucial in the establishment of the LeVeque type inequalities that are capable of giving dis- crepancy estimates for some minimal energy configurations. We provide an overview of the recent developments outlined above. In the final section we show that many results on the sphere can be generalised to other compact homogeneous manifolds. We also propose a few research topics for future investigation in this area.