32 research outputs found
Sampling of Entire Functions of Several Complex Variables on a Lattice and Multivariate Gabor Frames
We give a general construction of entire functions in complex variables
that vanish on a lattice of the form for an invertible
complex-valued matrix. As an application we exhibit a class of lattices of
density >1 that fail to be a sampling set for the Bargmann-Fock space in . By using an equivalent real-variable formulation, we show that these
lattices fail to generate a Gabor frame
Uncertainty Principles and Vector Quantization
Given a frame in C^n which satisfies a form of the uncertainty principle (as
introduced by Candes and Tao), it is shown how to quickly convert the frame
representation of every vector into a more robust Kashin's representation whose
coefficients all have the smallest possible dynamic range O(1/\sqrt{n}). The
information tends to spread evenly among these coefficients. As a consequence,
Kashin's representations have a great power for reduction of errors in their
coefficients, including coefficient losses and distortions.Comment: Final version, to appear in IEEE Trans. Information Theory.
Introduction updated, minor inaccuracies corrected
Radial oscillation of harmonic functions in the Korenblum class
We study radial behavior of harmonic functions in the unit disk belonging to
the Korenblum class. We prove that functions which admit two-sided Korenblum
estimate either oscillate or have slow growth along almost all radii
Phase space distribution of Gabor expansions
We present an example of a complete and minimal Gabor system consisting of
time-frequency shifts of a Gaussian, localized at the coordinate axes in the
time-frequency plane (phase space). Asymptotically, the number of
time-frequency shifts contained in a disk centered at the origin is only 2/pi
times the number of points from the von Neumann lattice found in the same disk.
Requiring a certain regular distribution in phase space, we show that our
system has minimal density among all complete and minimal systems of
time-frequency shifts of a Gaussian.Comment: 9 pages, submitted to publicatio