# Construction of Minimizers

The goal is to develop a construction scheme for minimizers of the uncertainty principles developed in UNLocX. By “minimizer” we mean optimal waveforms with respect to the generalized Localization Measures. This construction schemes will include and generalize the construction principles for minimizing Heisenberg type inequalities and/or inequalities derived from entropy or ambiguity function formulations. Moreover, we aim at developing a construction-scheme for uncertainty minimizers in view of specific signal processing tasks (e.g. signal coding and signal analysis/understanding). Thus we shall provide construction principles for optimal waveforms in the sense of the task-oriented versions of uncertainty inequalities developed in the component Localization Measures.

We have the following aims:

- Variational concept: For the uncertainty inequalities derived in Localization Measures we shall formulate and analyze associated variational principles. (Differential) equations, which have to be satisfied by minimizers of these variational principles will be given and solution schemes will be outlined. In order to reach this aim we identify several tasks, each of which addresses the derivation of individual variational principles associated to the individual uncertainty inequalities (Heisenberg-type, entropy based) mentioned above.
- Explicit construction of minimizers: If applicable, we shall develop explicit construction schemes for minimizers; this in particular will be important for constructing minimizers of task-oriented uncertainty inequalities.
- M-frame constructions: Variational methods and explicit construction schemes will be generalized to function systems with more than one elementary waveform. Thus construction schemes for obtaining combined minimizers for these waveforms will be given.