Discrete Hilbert transforms on sparse sequences
By Y. Belov, T. Mengestie, Kristian Seip. 2009
61 citations recorded by the source corpus.
Weighted discrete Hilbert transforms (an)n ↦ ∑n an vn/(z−γn) from ℓ2ν to a weighted L2‐space are studied, with Γ=(γn) a sequence of distinct points in the complex plane and v=(vn) a corresponding sequence of positive numbers. In the special case when |γn| grows at least exponentially, bounded transforms of this kind are described in terms of a simple relative to the Muckenhoupt (A2) 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 the invertibility of two such 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 H2 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.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/3b55b0ccd1edde361e2bf80e711701146edb729c
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In Applied Mathematics
Citations in the corpus
- Riesz bases of reproducing kernels in Fock-type spaces (2009, background): “Our work can be considered as completing a series of works [4], [2], and [1].”
- Hereditary completeness for systems of exponentials and reproducing kernels (2011, background): “At first glance it may look like an artificial trick; however it should be noted that the existence of two orthogonal bases of reproducing kernels is a property which characterizes de Branges spaces among all Hilbert spaces of entire functions (see [5,6]).”