Plancherel transform criteria for Weyl-Heisenberg frames with integer oversampling

We investigate the relevance of admissibility criteria based on Plancherel measure for the characterization of tight Weyl-Heisenberg frames with integer oversampling. For this purpose we observe that functions giving rise to such Weyl-Heisenberg frames are admissible with regard to the action of a suitably defined type-I discrete group G. This allows to relate the construction of Weyl-Heisenberg frames to the Plancherel measure of G, which provides an alternative proof and a new interpretation of the well-known Zak transform based criterion for tight Weyl-Heisenberg frames with integer oversampling. 1 Admissibility conditions and Weyl-Heisenberg frames This paper interprets characterizations of tight Weyl-Heisenberg frames as admissibility conditions connected to a certain discrete group. The starting point was an observed similarity between Zak transform based criteria for such frames and representation-theoretic admissibility conditions established by the author. We will show that the former can be seen as special instances of the latter. In order to review the notion of admissibility, let G be a locally compact group with left Haar measure μG. We let L 2(G) denote the associated L2space, on which G acts unitarily by left translations; this defines the left ∗email: [email protected]