  Weighted discrete Hilbert transforms $(a_n)_n \mapsto \big(\sum_n a_n
v_n/(\lambda_j-\gamma_n)\big)_j$ from $\ell^2_v$ to $\ell^2_w$ are considered,
where $\Gamma=(\gamma_n)$ and $\Lambda=(\lambda_j)$ are disjoint sequences of
points in the complex plane and $v=(v_n)$ and $w=(w_j)$ are positive weight
sequences. It is shown that if such a Hilbert transform is unitary, then
$\Gamma\cup\Lambda$ is a subset of a circle or a straight line, and a
description of all unitary discrete Hilbert transforms is then given. A
characterization of the orthogonal bases of reproducing kernels introduced by
L. de Branges and D. Clark is implicit in these results: If a Hilbert space of
complex-valued functions defined on a subset of $\CC$ satisfies a few basic
axioms and has more than one orthogonal basis of reproducing kernels, then
these bases are all of Clark's type.
