Properties of the voice transform of the Blaschke group and connections with atomic decomposition results in the weighted Bergman spaces

Feichtinger and Gröchenig described a unified approach to atomic decomposition through integrable group representations in Banach spaces. Studying the properties of a special voice transform of the Blascke group, outlined by the general theory developed by Feichtinger and Gröchenig, we obtain that every function from the minimal Möbius invariant space will generate an atomic decomposition in the weighted Bergman spaces. In the unified approach of the atomic decomposition a useful tool is the Q-density, the V-separated property and the bounded uniform partitions of the unity of the locally compact group. Using the hyperbolic metric we can describe the Q-density from right, and the separation from right in the Blaschke group. Using this we can give an example of bounded uniform partitions of the unity from right. In the general theory of atomic decomposition it is used the Q-density from the left, this is the reason why we will make a small modification in the discretizing operator which corresponds to the Q-density from the right in order to obtain atomic decomposition in the weighted Bergman spaces.