  We introduce a construction of multiscale tight frames on general domains.
The frame elements are obtained by spectral filtering of the integral operator
associated with a reproducing kernel. Our construction extends classical
wavelets as well as generalized wavelets on both continuous and discrete
non-Euclidean structures such as Riemannian manifolds and weighted graphs.
Moreover, it allows to study the relation between continuous and discrete
frames in a random sampling regime, where discrete frames can be seen as Monte
Carlo estimates of the continuous ones. Pairing spectral regularization with
learning theory, we show that a sample frame tends to its population
counterpart, and derive explicit finite-sample rates on spaces of Sobolev and
Besov regularity. Our results prove the stability of frames constructed on
empirical data, in the sense that all stochastic discretizations have the same
underlying limit regardless of the set of initial training samples.
