Affine density, von Neumann dimension and a problem of Perelomov.

We provide a solution to Perelomov's 1972 problem concerning the existence of a phase transition (known in signal analysis as 'Nyquist rate') determining the basis properties of certain affine coherent states labelled by Fuchsian groups. As suggested by Perelomov, the transition is given according to the hyperbolic volume of the fundamental region. The solution is a more general form (in phase space) of the $PSL(2,\mathbb{R})$ variant of a 1989 conjecture of Kristian Seip about wavelet frames, where the same value of `Nyquist rate' is obtained as the trace of a certain localization operator. The proof consists of first connecting the problem to the theory of von Neumann algebras, by introducing a new class of projective representations of $PSL(2,\mathbb{R})$ acting on non-analytic Bergman-type spaces. We then adapt to this setting a new method for computing von Neumann dimensions, due to Sir Vaughan Jones. Our solution contains necessary conditions in the form of a `Nyquist rate' dividing frames from Riesz sequences of coherent states and sampling from interpolating sequences. They hold for an infinite sequence of spaces of polyanalytic functions containing the eigenspaces of the Maass operator and their orthogonal sums. Within mild boundaries, we show that our result is best possible, by characterizing our sequence of function spaces as the only invariant spaces under the non-analytic $PSL(2,\mathbb{R})$-representations.