Coorbit Space Theory for Quasi-Banach Spaces

We generalize the classical coorbit space theory developed by Feichtinger and Gr"ochenig to quasi-Banach spaces. As a main result we provide atomic decompositions for coorbit spaces defined with respect to quasi-Banach spaces. These atomic decompositions are used to prove fast convergence rates of best $n$-term approximation schemes. We apply the abstract theory to time-frequency analysis of modulation spaces $M_m^{p,q}$, $0<p,q\le \mathrm{\infty }$.