Unitary discrete Hilbert transforms

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 $\mathrm{\Gamma }=({\gamma }_n)$ and $\mathrm{\Lambda }=({\lambda }_j)$ 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 $\mathrm{\Gamma }\cup \mathrm{\Lambda }$ 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 $\text{CC}$ satisfies a few basic axioms and has more than one orthogonal basis of reproducing kernels, then these bases are all of Clark’s type.