Boundedness and invertibility for disrete Hilbert transform with sparse poles

We are interested in the following question:
For which $v_n$ and $\mu$ a discrete Hilbert transform $H((a_n)) = \sum_n a_n*v_n/(z-t_n)$ is a bounded operator from $l^2(v_n)$ to $L^2(d\mu,C)$? For a fast growing $|t_n|$ we give necessary and sufficient conditions. These conditions are similar to a classical Muckenhoupt condition. Discrete Hilbert transform naturally appears in studies of spaces of entire functions with Riesz basis from reproducing kernels (Paley-Wiener spaces, de Branges spaces, weighted Fock spaces e.t.c.). In particular our results make it posssible to give a characterization of all Carleson measures (Bessel sequences) and all Riesz basis in "small" spaces of entire functions as well we will check Feichtinger hypothesis for such spaces (and reproducing kernels). (Joint work with K. Seip and T. Mengestie)