Time-Frequency generating systems using model space methods

Generating systems are sets of vectors that span the space that are under consideration. These systems give a description of the vectors of the space. This project deals with systems that have good time and frequency representation properties for the L2(R) space and for the modulation spaces. These systems are composed by translations, modulations or/and dilations of a fixed function. Gabor and wavelet frames are examples of such systems. The problems proposed are addressed to the construction of systems of this type, and their properties. Most of the interest will go to irregular Gabor and wavelet frames, their localization properties and the possibility to extend the knowledge of these systems in L2(R) to the family of modulation spaces. Generation by translations and alpha-modulations systems will also be considered. The project wants to consider this problem from a unified point of view, using continuous transforms and considering sampling and uniqueness problems on certain spaces of functions.