Discrepancy kernels on manifolds and their Fourier coefficients
given at NuHAG seminar (14.03.18 11:00)
The approximation of continuous objects with discrete ones is a fundamental problem in applied mathematics. The various notions of discrepancy in the literature enable quantifying the difference or similarity of continuous and discrete measures. In particular, minimizing discrepancy kernels provide a means to approximate continuous measures by discrete ones.
Many problems in the applied sciences yield data that lie on manifolds, leading to discrepancy problems on manifolds. Here, we study families of discrepancy kernels on closed Riemannian manifolds. First, we represent the discrepancy kernel by means of a function of the distance. This new representation enables us to specify the reproducing kernel Hilbert space of the discrepancy kernel as a particular Sobolev space. Our results generalize findings by Brauchart and Dick from the unit sphere to more general spaces.