The geometry of sets of parameters of wave packet frames (Q818337)

A wave packet system consists of a countable collection of dilated, modulated, and translated versions of a function \(\psi\in L^2(\mathbb R)\). The parameters of these operations form a discrete subset of \(\mathbb R^+ \times \mathbb R\times\mathbb R\); adapted to such subsets, the authors introduce a notion of density, and associated upper and lower dimension. Sufficient conditions for a wave packet system to form a Bessel sequence are found, and it is observed that the so-called regular systems usually are not Bessel sequences. On the other hand, some irregular wave packet systems which form Bessel sequences (they are even frames) are constructed.