"Dual molecules for (quasi-Banach) coorbit spaces"Voigtlaender, FelixThis talk is concerned with discretization in coorbit spaces defined with respect to a unitary group representation. Examples of such coorbit spaces include the modulation spaces and (possibly anisotropic) Besov spaces, as well as a wide class of shearlet coorbit spaces. Classical coorbit theory shows that if the generating vector (e.g., a window function or a mother wavelet) is "nice enough", then one can sample the orbit of this generating vector under the group representation (e.g., a continuous Gabor or wavelet system) to find a countable sub-system that forms an atomic decomposition and a Banach frame for the coorbit spaces in question. Among other things, this implies that elements of suitable coorbit spaces are well approximated by linear combinations of only a few elements of this discrete system. In this talk, we strengthen this discretization result by showing that the dual family associated to the discretized system can be chosen in such a way that it satisfies strong localization properties. These results strengthen the conclusions of classical coorbit theory, while at the same time producing simplified proofs. Moreover, the results apply in great generality, namely in the setting of quasi-Banach spaces and for possibly reducible, projective representations. The proof is based on a certain local spectral invariance for convolution-dominated integral operators. This is joint work with Jordy van Velthoven, based on further joint work with Jose Luis Romero and Jordy van Velthoven. |
« back