International Conference on Multivariate Approximation
September 24-27, 2011

Kernel based Methods and Radial Basis Functions

Positive definite kernels and radial basis functions provide the mathematical foundation for several methods of multivariate approximation. Shifts of a given kernel provide well-approximating trial spaces, for interpolation, approximation, or discretization of partial differential equations. One of the key features of positive definite kernels is that the data-driven selection of shifts, originating from scattered (or meshless) data, provides a stable basis of a subspace with respect to a "natural" norm. As properties of the positive definite kernel determine this norm, the kernel can be designed according to the desired goals of the approximation method. The design of positive definite kernels is also a crucial issue in statistical learning theory and kernel learning.