Energy Blowup of Sampling-based Approximation Methods

This paper considers the problem of approximating continuous functions of finite Dirichlet energy from samples of these functions. It will be shown that there exists no sampling-based method which is able to approximate every function in this space from its samples. Specifically, we are going to show that for any sampling based approximation method, the energy of the approximation tends to infinity as the number of samples is increased for almost every continuous function of finite energy. As an application, we study the problem of solving the Dirichlet problem on a bounded region. It will be shown that if only samples of the boundary function can be processed then the energy of the solution can not be controlled for any function from a non-meager dense set.