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On homogeneous decomposition spaces and associated decompositions of distribution spaces

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Abstract 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 Rd∖{0}. We construct simple adapted tight frames for L2(Rd) 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 α‐modulation spaces is introduced.

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