Affine density and von Neumann dimension

Adapting a method to compute the von Neumann dimension of a factor, recently presented by Sir Vaughan Jones, we provide a solution to an old problem about basis properties of affine coherent states (also known as analytic wavelets) labelled by Fuchsian groups, which can be traced back to the work of Perelomov in the early 70's. The solution contains the description of sampling and interpolating sequences in eigenspaces of the Maass operator, in the form of a 'Nyquist rate' dividing frames and Riesz sequences for a certain orthonormal family of wavelet functions, as typical of such problems for signal representations. The value of the `Nyquist rate' coincides with a conjecture of Kristian Seip in 1989.