General superconvergence for kernel-based approximation

Kernel interpolation is a fundamental technique for approximating functions from scattered data, with a well-understood convergence theory when interpolating elements of a reproducing kernel Hilbert space. Beyond this classical setting, research has focused on two regimes: misspecified interpolation, where the kernel smoothness exceeds that of the target function, and superconvergence, where the target is smoother than the Hilbert space. This work addresses the latter, where smoother target functions yield improved convergence rates, and extends existing results by characterizing superconvergence for projections in general Hilbert spaces. We show that functions lying in ranges of certain operators, including adjoint of embeddings, exhibit accelerated convergence, which we extend across interpolation scales between these ranges and the full Hilbert space. In particular, we analyze Mercer operators and embeddings into $L_p$ spaces, linking the images of adjoint operators to Mercer power spaces. Applications to Sobolev spaces are discussed in detail, highlighting how superconvergence depends critically on boundary conditions. Our findings generalize and refine previous results, offering a broader framework for understanding and exploiting superconvergence. The results are supported by numerical experiments.