Square integrable representations, von Neumann algebras and an application to Gabor analysis (Q1882377)

The author studies square integrable representations \(\pi:G\to H_\pi\) of a locally compact group \(G\) and the restriction of this representation to a discrete subgroup, in particular, the question when the restriction has a cyclic vector. Since this restriction extends to a representation of a certain von Neumann algebra, techniques and concepts from these can be used. The author determines the center valued von Neumann dimension of \(H_\pi\) and explicit formulas are given in the case where \(G\) is a semisimple algebraic group or a nilpotent Lie group. As an application, it is shown that if \(\Lambda\) is a lattice in \(\mathbb R^{2d}\), then one can choose a function \(g\) so that the functions \(g^{\omega,t} = e^{-i\omega \cdot s} \overline{g(s-t)}\), \((\omega ,t)\in \Lambda\) (Gabor wavelets) form a complete set in \(L^2(\mathbb R^d)\) if and only if vol\((\mathbb R^{2d}/\Lambda) \leq 2\pi\).