Discrete Hilbert transforms on sparse sequences

Weighted discrete Hilbert transforms (a(n))(n) & Sigma(n) a(n) v(n)/(z-gamma(n)) from l(nu)(2) to a weighted L(2)-space are studied, with Gamma=(gamma(n)) a sequence of distinct points in the complex plane and v=(v(n)) a corresponding sequence of positive numbers. In the special case when vertical bar gamma(n)vertical bar grows at least exponentially, bounded transforms of this kind are described in terms of a simple relative to the Muckenhoupt (A(2)) condition. The special case when z is restricted to another sequence Lambda is studied in detail; it is shown that a bounded transform satisfying a certain admissibility condition can be split into finitely many surjective transforms, and precise geometric conditions are found for the invertibility of two such weight transforms. These results can be interpreted as statements about systems of reproducing kernels in certain Hilbert spaces of which de Branges spaces and model subspaces of H(2) are prime examples. In particular, a connection to the Feichtinger conjecture is pointed out. Descriptions of Carleson measures and Riesz bases of normalized reproducing kernels for certain 'small' de Branges spaces follow from the results of this paper.