Tesfa Y. Mengestie | Searcharxiv

arXiv2010

Unitary discrete Hilbert transforms

Weighted discrete Hilbert transforms $(a_n)_n \mapsto \big(\sum_n a_n v_n/(λ_j-γ_n)\big)_j$ from $\ell^2_v$ to $\ell^2_w$ are considered, where $Γ=(γ_n)$ and $Λ=(λ_j)$ 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 $Γ\cupΛ$ 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.

math.CV↗

arXiv2009

Discrete Hilbert transforms on sparse sequences

Weighted discrete Hilbert transforms $(a_n)_n \mapsto \sum_n a_n v_n/(z-γ_n)$ from $\ell^2_v$ to a weighted $L^2$ space are studied, with $Γ=(γ_n)$ a sequence of distinct points in the complex plane and $v=(v_n)$ a corresponding sequence of positive numbers. In the special case when $|γ_n|$ grows at least exponentially, bounded transforms of this kind are described in terms of a simple relative to the Muckenhoupt $(A_2)$ condition. The special case when $z$ is restricted to another sequence $Λ$ is studied in detail; it is shown that a bounded transform satisfying a certain admissibility condition can be split into finitely many surjective transforms, and precise geometric conditions are found for invertibility of such two weight transforms. These results can be interpreted as statements about systems of reproducing kernels in certain Hilbert spaces of which de Branges spaces and model subspaces of $H^2$ are prime examples. In particular, a connection to the Feichtinger conjecture is pointed out. Descriptions of Carleson measures and Riesz bases of normalized reproducing kernels for certain "small" de Branges spaces follow from the results of this paper.

math.CV↗