Banach frames in coorbit spaces consisting of elements which are invariant under symmetry groups

This paper is concerned with the construction of atomic decompositions and Banach frames for subspaces of certain Banach spaces consisting of elements which are invariant under some symmetry group. These Banach spaces-called coorbit spaces-are related to an integrable group representation. The construction is established via a generalization of the well-established Feichtinger-Gröchenig theory. Examples include radial wavelet-like atomic decompositions and frames for radial Besov-Triebel-Lizorkin spaces, as well as radial Gabor frames and atomic decompositions for radial modulation spaces.