Unitary discrete Hilbert transforms

Abstract

Weighted discrete Hilbert transforms $(a_n{)}_n\mapsto ({\sum }_n{a}_n{v}_n/({\lambda }_j-{\gamma }_n){)}_j$ from ${\ell }_v^2$ to ${\ell }_w^2$ 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.

Cited by in corpus (2)

• Fredholmness and compactness of truncated Toeplitz and Hankel operators
• One-parameter groups of operators and discrete Hilbert transforms