Admissible vectors, density conditions and localization

Coherent states provide a powerful method to decompose functions into components of a simpler form. This method is commonly known as an atomic decomposition. It allows to study the action of operators on functions by first considering the operation on the components, so-called atoms. The atoms arise from a single action of a unitary representation of a Lie group. A remarkable property of atoms or states is that they are non-orthogonal and overcomplete, so the system contains more elements than necessary to decompose a function. If treated by means of proper methods such as frame theory, it is precisely this property which opened up the spectacular applications of coherent states for the simplest Lie groups in areas of physics (quantum theory), mathematics (harmonic and functional analysis) and engineering (signal analysis). The aim of this project is to study the spanning properties of subsystems of coherent states for classes of representations and Lie groups. A particular focus is on density conditions for frames and Riesz sequences with localized vectors. Such conditions provide criteria for the completeness of the subsystem and are formulated in terms of a notion of density of the index set. The challenge in this is to determine the critical value of density at which the subsystem forms both a frame and a Riesz sequence, a so-called Riesz basis. The incommensurability between critical density and localization of the system often leads to strong types of uncertainty principles for Riesz bases. The techniques that will be investigated are from diverse fields of mathematics (harmonic analysis, operator theory and Lie theory) and mathematical physics. The tools that will be exploited are structural results that are significant for the classes of representations and Lie groups under consideration.

Decomposing functions into basic components is a powerful technique for the analysis of functions and operators acting on them. This is because the basic components (often called atoms) possess a particularly simple form on which the action of the operator is relatively easy to understand. Classical examples of such decompositions are Fourier series of periodic functions and atomic decompositions of Hardy spaces. The project studied functional expansions in which the atoms are derived from a single function (often called the template) by means of a group action. Atoms of this particular form have been studied and used in various areas of mathematical analysis and physics. During the project, it has been investigated under which localisation conditions on the template functional expansions can be obtained and which critical value of density the index set of the atoms needs to satisfy for such expansions to hold. In addition, the question whether functional expansions with well-localised template at the critical density are possible has been investigated. The techniques used during the project stem from various areas of functional analysis, harmonic analysis and representation theory. More specifically, these questions have been addressed through the notions of Beurling density, frames and Riesz bases and square-integrable group representations. Using these notions, the project has unravelled new phenomena for groups with exponential growth and for index sets not forming a subgroup. In addition to the questions of density and localisation of atoms for functional expansions, the project studied function spaces generated by atoms obtained from translations and anisotropic dilations. For such spaces, complete classifications have been obtained for the dependence of the spaces on the anisotropic dilation matrix.