Unitary discrete Hilbert transforms

TL;DR

This paper characterizes when weighted discrete Hilbert transforms are unitary, showing they occur only when point sequences lie on a circle or line, and links these results to orthogonal bases of reproducing kernels.

Contribution

It provides a complete characterization of unitary discrete Hilbert transforms and connects these transforms to the structure of orthogonal bases of reproducing kernels.

Findings

Unitary discrete Hilbert transforms occur only when point sequences are on a circle or line.

All orthogonal bases of reproducing kernels in certain function spaces are of Clark's type.

The paper offers a classification of unitary transforms based on geometric configurations.

Abstract

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.