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

Construction and Monte Carlo estimation of wavelet frames generated by a reproducing kernel