WP2: Harmonic analysis and approximation theory

We will obtain a full noncommutative theory of sampling in group invariant Hilbert subspaces. This will represent a huge theoretical novelty, as it will bring in the full power of noncommutative Fourier analysis, and it will open many applications of data analysis in the presence of invariances. We will also address problems of reproducing kernel Hilbert spaces of several complex variables, which are strictly related to functional calculus and contractions in operator theory, focusing on the characterization of the multiplier algebras and of the structure of interpolating sequences. This will lead to the development of new tools of potential theory, which will also be applied to the special cases of invariant reproducing kernel Hilbert spaces generated by unitary representations of Lie groups that are not necessarily square integrable. These will require the study of restrictions of the group Fourier transform to homogeneous manifolds, and their complex regularity.