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Kernel interpolation is a versatile tool for the approximation of functionsfrom data, and it can be proven to have some optimality properties when usedwith kernels related to certain Sobolev spaces.  In the context of interpolation, the selection of optimal function samplinglocations is a central problem, both from a practical perspective, and as aninteresting theoretical question. Greedy interpolation algorithms provide aviable solution for this task, being efficient to run and provably accurate intheir approximation.  In this paper we close a gap that is present in the convergence theory forthese algorithms by employing a recent result on general greedy algorithms.This modification leads to new convergence rates which match the optimal oneswhen restricted to the $P$-greedy target-data-independent selection rule, andcan additionally be proven to be optimal when they fully exploit adaptivity($f$-greedy).  Other than closing this gap, the new results have some significance in thebroader setting of the optimality of general approximation algorithms inReproducing Kernel Hilbert Spaces, as they allow us to compare adaptiveinterpolation with non-adaptive best nonlinear approximation.

On the optimality of target-data-dependent kernel greedy interpolation in Sobolev Reproducing Kernel Hilbert Spaces