In mathematics, coorbit theory was developed by Hans Georg Feichtinger and Karlheinz Gröchenig around 1990.
[1][2][3] It provides theory for atomic decomposition of a range of Banach spaces of distributions.
The starting point is a square integrable representation
Representation theory yields the reproducing formula
By discretization of this continuous convolution integral it can be shown that by sufficiently dense sampling in phase space the corresponding functions will span a frame for the Hilbert space.
An important aspect of the theory is the derivation of atomic decompositions for Banach spaces.
One of the key steps is to define the voice transform for distributions in a natural way.
The reproducing formula is true also in this case and therefore it is possible to obtain atomic decompositions for coorbit spaces.