Yurii Belov

Unitary discrete Hilbert transforms

AbstractWeighted discrete Hilbert transforms

Summability properties of Gabor expansions

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

The Young type theorem in weighted Fock spaces

from ℓυ2 to ℓw2 are considered, where Γ = (γn) and ∧ = (λj) are disjoint sequences of points in the complex plane and υ = (υn) and ω = (ωj) 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.

DISCRETE HILBERT TRANSFORMS ON SPARSE SEQUENCES

We study the localization of zeros of Cauchy transforms of discrete measures on the real line. This question is motivated by the theory of canonical systems of differential equations. In particular, we prove that the spaces of Cauchy transforms having the localization property are in one-to-one correspondence with the canonical systems of special type, namely, those whose Hamiltonians consist only of indivisible intervals accumulating on the left. Various aspects of the localization phenomena are studied in details. Connections with the density of polynomials and other topics in analysis are discussed.

Hereditary completeness for systems of exponentials and reproducing kernels

Abstract We show that there exist complete and minimal systems of time-frequency shifts of Gaussians in L 2 ( R ) which are not strong Markushevich basis (do not admit the spectral synthesis). In particular, it implies that there is no linear summation method for general Gaussian Gabor expansions. On the other hand we prove that the spectral synthesis for such Gabor systems holds up to one dimensional defect.

Systems of Reproducing Kernels and their Biorthogonal: Completeness or Incompleteness?

Abstract We solve a problem about the orthogonal complement of the space spanned by restricted shifts of functions in L 2 [ 0 , 1 ] posed by M. Carlsson and C. Sundberg.

Spectral synthesis in de Branges spaces

We prove that for every radial weighted Fock space, the system biorthogonal to a complete and minimal system of reproducing kernels is also complete under very mild regularity assumptions on the weight. This result generalizes a theorem by Young on reproducing kernels in the Paley-Wiener space and a recent result of Belov for the classical Bargmann-Segal-Fock space.