Frame Decomposition of Decomposition Spaces

Abstract

A new construction of tight frames for $L_2({\Bbb R}^d)$ with flexible time-frequency localization is considered. The frames can be adapted to form atomic decompositions for a large family of smoothness spaces on ${\Bbb R}^d,$ 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, α-modulation spaces, and anisotropic α-modulation spaces are considered. Finally, curvelet-type tight frames are constructed on ${\Bbb R}^d, d \geq 2.$