On Homogeneous Decomposition Spaces and Associated Decompositions of Distribution Spaces

A new construction of decomposition smoothness spaces of homogeneous type is considered. The smoothness spaces are based on structured and flexible decompositions of the frequency space $\mathbb{R}^d\backslash\{0\}$. We construct simple adapted tight frames for $L_2(\mathbb{R}^d)$ that can be used to fully characterise the smoothness norm in terms of a sparseness condition imposed on the frame coefficients. Moreover, it is proved that the frames provide a universal decomposition of tempered distributions with convergence in the tempered distributions modulo polynomials. As an application of the general theory, the notion of homogeneous $\alpha$-modulation spaces is introduced.