Approximation in kernel-based spaces, optimal subspaces and approximation of eigenfunctions

Kernel-based approximation methods provide optimal recovery procedures in the native Hilbert spaces in which they are reproducing. Among other, kernels in the notable class of continuous and strictly positive definite kernels on compact sets possess a series decomposition in L2 - orthonormal eigenfunctions of a particular integral operator. The interest for this decomposition is twofold. On one hand, the subspaces generated by eigenfunctions, or eigenbasis elements, are L2 -optimal trial spaces in the sense of widths. On the other hand, such expansion is the fundamental tool of some of the state of the art algorithms in kernel approximation. Despite these reasons motivate a great interest in the eigenbasis, for a given kernel this decomposition is generally completely unknown. In this view, this thesis faces the problem of approximating the eigenbasis of general continuous and strictly positive definite kernels on general compact sets of the Euclidean space, for any space dimension. We will at first define a new kind of optimality that is based on a error measurement closer to the one of standard kernel interpolation. This new width is then analyzed, and we will determine its value and characterize its optimal subspaces, which are spanned by the eigenbasis. Moreover, this optimality result is suitable to be scaled to some particular subspace of the native space, and this restriction allows us to prove new results on the construction of computable optimal trial spaces. This situation covers the standard case of point-based interpolation, and will provide algorithms to approximate the eigenbasis by means of standard kernel techniques. On the basis of the new theoretical results, asymptotic estimates on the convergence of the method will be proven. These computations will be translated into effective algorithms, and we will test their behavior in the approximation of the eigenspaces. Moreover, two applications of kernel-based methods will be analyzed