1,651 research outputs found
The Newman--Shapiro problem
We give a negative answer to the Newman--Shapiro problem on weighted
approximation for entire functions formulated in 1966 and motivated by the
theory of operators on the Fock space. There exists a function in the Fock
space such that its exponential multiples do not approximate some entire
multiples in the space. Furthermore, we establish several positive results
under different restrictions on the function in question.Comment: 28 page
Hereditary completeness for systems of exponentials and reproducing kernels
We solve the spectral synthesis problem for exponential systems on an
interval. Namely, we prove that any complete and minimal system of exponentials
in is hereditarily complete up to a
one-dimensional defect. This means that there is at most one (up to a constant
factor) function which is orthogonal to all the summands in its formal
Fourier series , where
is the system biorthogonal to . However, this
one-dimensional defect is possible and, thus, there exist nonhereditarily
complete exponential systems. Analogous results are obtained for systems of
reproducing kernels in de Branges spaces. For a wide class of de Branges spaces
we construct nonhereditarily complete systems of reproducing kernels, thus
answering a question posed by N. Nikolski.Comment: 35 pages. Major changes in Sections 4 and 5. An example of a
nonhereditarily complete system of exponentials is constructe
The Young type theorem in weighted Fock 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.Comment: 7 page
Nevanlinna domains with large boundaries
We establish the existence of Nevanlinna domains with large boundaries. In
particular, these domains can have boundaries of positive planar measure. The
sets of accessible points can be of any Hausdorff dimension between and
. As a quantitative counterpart of these results, we construct rational
functions univalent in the unit disc with extremely long boundaries for a given
amount of poles
Summability properties of Gabor expansions
We show that there exist complete and minimal systems of time-frequency
shifts of Gaussians in 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.Comment: 21 page
Spectral synthesis in de Branges spaces
We solve completely the spectral synthesis problem for reproducing kernels in
the de Branges spaces . Namely, we describe the de Branges
spaces such that all -bases of reproducing kernels (i.e.,
complete and minimal systems with complete
biorthogonal ) are strong -bases (i.e.,
every mixed system is also complete). Surprisingly
this property takes place only for two essentially different classes of de
Branges spaces: spaces with finite spectral measure and spaces which are
isomorphic to Fock-type spaces of entire functions. The first class goes back
to de Branges himself, the second class appeared in a recent work of A.
Borichev and Yu. Lyubarskii. Moreover, we are able to give a complete
characterisation of this second class in terms of the spectral data for
. In addition, we obtain some results about possible
codimension of mixed systems for a fixed de Branges space , and
prove that any minimal system of reproducing kernels in is
contained in an exact system of reproducing kernels.Comment: 38 pages. Shortened text with streamlined proofs. This version is
accepted for publication in "Geometric and Functional Analysis
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