  Besides standard Lagrange interpolation, i.e., interpolation of target
functions from scattered point evaluations, positive definite kernel functions
are well-suited for the solution of more general reconstruction problems. This
is due to the intrinsic structure of the underlying reproducing kernel Hilbert
space (RKHS). In fact, kernel-based interpolation has been applied to the
reconstruction of bivariate functions from scattered Radon samples in
computerized tomography (cf. Iske, 2018) and, moreover, to the numerical
solution of elliptic PDEs (cf. Wenzel et al., 2022). As shown in various
previous contributions, numerical algorithms and theoretical results from
kernel-based Lagrange interpolation can be transferred to more general
interpolation problems. In particular, greedy point selection methods were
studied in (Wenzel et al., 2022), for the special case of Sobolev kernels. In
this paper, we aim to develop and analyze more general kernel-based
interpolation methods, for less restrictive settings. To this end, we first
provide convergence results for generalized interpolation under minimalistic
assumptions on both the selected kernel and the target function. Finally, we
prove convergence of popular greedy data selection algorithms for totally
bounded sets of functionals. Supporting numerical results are provided for
illustration.
