Page 39, proposition 1.1.3 here, http://www.cis.upenn.edu/~cis610/sharmonics.pdf clearly explains how for every ``level" (the parameter $k$ in the proposition) one can construct a function ("kernel") such that the function space of all level $k$ harmonics on $S^n$ is the "Reproducing Kernel Hilbert Space" (RKHS) for that kernel. Because of the orthogonality property of the harmonics it probably wasn't surprising that the kernel itself also turned out to be completely described by the harmonics themselves at the same level.

Firstly I want to confirm if my above interpretation is correct at all! I am inquiring if this proposition 1.1.3 in my cited reference can at all be see as constructing a RKHS. Because as far as I understand for their $F_k$ function in the reference to be considered as a "kernel" it has to converge pointwise and also maybe uniformly on compact sets. Is that true here?

Is this something special about $S^n$ or are there other (compact?) manifolds too on which the Hilbert space of harmonic functions similarly decomposes so that the eigenspace of each eigenvalue acts as a RKHS for the kernel being constructed similarly? (and this kernel will again have the "right" (pointwise? uniformly on compact sets?) convergent properties?)

wholespace of "spherical harmonics", but only the finite-dimensional space of those of a fixed degree, which is to say those with a fixed eigenvalue for the invariant Laplacian on the sphere. On every compact Riemannian manifold, the eigenspaces for the Laplace-Beltrami operator are finite-dimensional, so the identity map (whose "kernel" is the "reproducing kernel") is a nice function of two variables... yes. :) ? $\endgroup$ – paul garrett May 4 '16 at 22:21NORin $L^2$. The idea of a reproducing kernel is that it converges pointwise to a considerable degree... $\endgroup$ – paul garrett May 4 '16 at 22:37allspherical harmonics, is definitely not convergent. So, while there are indeed "nice properties", their proper description is not so trivial. $\endgroup$ – paul garrett May 4 '16 at 23:12