## Final Activity Report Summary - AGWPS (Analysis and Geometry of Wave Packet Systems)

We have systematically studied wave packets, that is systems of functions generated from a single window function, or from a finite collection of such windows, by the action of a countable family of dilations, modulations, and translations. We have done so by using the concept of Beurling density to introduce new notions of dimensions for countable subsets of locally compact Abelian groups. These new dimensions are the Beurling upper and lower dimensions. We have implemented these new notions of dimensions in a specific setting of the affine Weyl-Heisenberg group, which allowed us to provide a full characterisation of frames of wave packets with a shift invariant property. This characterisation takes the form of bounds for all the possible values of the Beurling dimensions of sets of parameters of discrete wave packets. The shift invariant property of wave packets is manifested by the assumption that the sets of parameters contain as a factor the integer lattice of modulations. This assumption can be weakened by the perturbation results, which allow us to modify the parameters of modulations without destroying the representation property of a wave packet frame. Moreover, we have introduced and studied the notion of equivalence of wave packets with respect to a pair of Banach spaces: a function space and a sequence space.

During the course of our studies we have constructed families of wave packets with different properties. These properties include constructions of wave packets with all possible allowed Beurling dimensions and simple generating windows, as well as constructions of wave packets with very regular windows, and sets of parameters which are subsets of smooth surfaces in the affine Weyl-Heisenberg group.

Our research lead to additional questions and new directions of study. The most important and promising line of investigation concerns the question of existence of uncertainty principles for orthonormal bases of wave packets. Such uncertainty principles exist for some special cases of wave packets, i.e., for Gabor systems or wavelets. They take the form of the so-called Balian-Low Theorem, which states that such bases cannot be constructed from window functions which are too well localised in the time-frequency domain (phase space). We have introduced several generalisations of these results and we studied the limitations which they pose for representation systems.

During the course of our studies we have constructed families of wave packets with different properties. These properties include constructions of wave packets with all possible allowed Beurling dimensions and simple generating windows, as well as constructions of wave packets with very regular windows, and sets of parameters which are subsets of smooth surfaces in the affine Weyl-Heisenberg group.

Our research lead to additional questions and new directions of study. The most important and promising line of investigation concerns the question of existence of uncertainty principles for orthonormal bases of wave packets. Such uncertainty principles exist for some special cases of wave packets, i.e., for Gabor systems or wavelets. They take the form of the so-called Balian-Low Theorem, which states that such bases cannot be constructed from window functions which are too well localised in the time-frequency domain (phase space). We have introduced several generalisations of these results and we studied the limitations which they pose for representation systems.