Coorbit theory

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 ${\displaystyle \pi }$ of a locally compact group ${\displaystyle {\mathcal {G}}}$ on a Hilbert space ${\displaystyle {\mathcal {H}}}$, with which one can define a transform of a function ${\displaystyle f\in {\mathcal {H}}}$ with respect to ${\displaystyle g\in {\mathcal {H}}}$ by ${\displaystyle V_{g}f(x)=\langle f,\pi (x)g\rangle }$. 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 ${\displaystyle V_{g}f=V_{g}f*V_{g}g}$. 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.

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