On the completeness of a system of coherent states

Completeness is proved for some subsystems of a system of coherent states. The liaear dependence of states is investigated for von Neumann type subsystems. A detailed study is made of the case when a regular lattice on the complex $\alpha$ plane with cell area $S=\pi$ corresponds to the states of the system. It is shown that in this case there exists only one linear relationship between the coherent states. This relationship is equivalent to an infinite set of identities, of which the simplest can also be obtained by means of the transformation formulas for $\theta$ functions.