In mathematics, coorbit theory was developed by Hans Georg Feichtinger and Karlheinz Gröchenig around 1990. It provides theory for atomic decomposition of a range of Banach spaces of distributions. Among others the well established wavelet transform and the short-time Fourier transform are covered by the theory.
The starting point is a square integrable representation formula_1 of a locally compact group formula_2 on a Hilbert space formula_3, with which one can define a transform of a function formula_4 with respect to formula_5 by formula_6. Many important transforms are special cases of the transform, e.g. the short-time Fourier transform and the wavelet transform for the Heisenberg group and the affine group respectively. Representation theory yields the reproducing formula formula_7. 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. For a given Banach space formula_8, the corresponding coorbit space is defined as the set of all distributions such that formula_9. The reproducing formula is true also in this case and therefore it is possible to obtain atomic decompositions for coorbit spaces.
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