Kernel approximation in reproducing kernel Hilbert spaces

Summary

Kernel approximation reconstructs a function from its values at finitely many points $x_j$. It uses a combination $\sum_j c_j K(\cdot, x_j)$ of copies of a positive definite kernel $K$. The natural setting is the reproducing kernel Hilbert space $H_K$ of the kernel. There the kernel interpolant is the element of smallest norm that matches the data. Its pointwise error is bounded by a power function times the norm of the target. It covers radial basis function interpolation and works in many variables and on scattered points.