Unitary discrete Hilbert transforms

Abstract

Weighted discrete Hilbert transforms

$$ (a_n )_n \mapsto \left( {\sum\limits_n {a_n } \upsilon _n /(\lambda _j - \gamma _n )} \right)_j $$

from ℓ ² _υ to ℓ ² _w are considered, where Γ = (γ _n ) and ∧ = (λ _j ) are disjoint sequences of points in the complex plane and υ = (υ _n ) and ω = (ω _j ) are positive weight sequences. It is shown that if such a Hilbert transform is unitary, then Γ ∪ Λ 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 ℂ satisfies a few basic axioms and has more than one orthogonal basis of reproducing kernels, then these bases are all of Clark’s type.