Frame decomposition of decomposition spaces

A new construction of tight frames for L-2 (R-d) with flexible time-frequency localization is considered. The frames can be adapted to form atomic decompositions for a large family of smoothness spaces on R-d, a class of so-called decomposition spaces. The decomposition space norm can be completely characterized by a sparseness condition on the frame coefficients. As examples of the general construction, new tight frames yielding decompositions of Besov space, anisotropic Besov spaces, alpha-modulation spaces, and anisotropic alpha-modulation spaces are considered. Finally, curvelet-type tight frames are constructed on R-d, d >= 2.