82 research outputs found
Counterexamples to the B-spline conjecture for Gabor frames
The frame set conjecture for B-splines , , states that the
frame set is the maximal set that avoids the known obstructions. We show that
any hyperbola of the form , where is a rational number smaller than
one and and denote the sampling and modulation rates, respectively, has
infinitely many pieces, located around , \emph{not} belonging to
the frame set of the th order B-spline. This, in turn, disproves the frame
set conjecture for B-splines. On the other hand, we uncover a new region
belonging to the frame set for B-splines , .Comment: Version 2: Lem. 5, Prop. 6, and Thm. 7 added, Version 3: Thm. 8
change
On the non-frame property of Gabor systems with Hermite generators and the frame set conjecture
The frame set conjecture for Hermite functions formulated in [Gr\"ochenig, J.
Fourier Anal. Appl., 20(4):865-895, 2014] states that the Gabor frame set for
these generators is the largest possible, that is, the time-frequency shifts of
the Hermite functions associated with sampling rates and modulation
rates that avoid all known obstructions lead to Gabor frames for
. By results in [Seip and Wallst\'en, J. Reine Angew. Math.,
429:107-113, 1992] and [Lemvig, Monatsh. Math., 182(4):899-912, 2017], it is
known that the conjecture is true for the Gaussian, the th order Hermite
functions, and false for Hermite functions of order ,
respectively. In this paper we disprove the remaining cases except for the
st order Hermite function
On some Hermite series identities and their applications to Gabor analysis
We prove some infinite series identities for the Hermite functions. From
these identities we disprove the Gabor frame set conjecture for Hermite
functions of order and for . The
results hold not only for Hermite functions, but for two large classes of
eigenfunctions of the Fourier transform associated with the eigenvalues
and , and the results indicate that the Gabor frame set of all such
functions must have a rather complicated structure
A Short Note on the Frame Set of Odd Functions
In this work we derive a simple argument which shows that Gabor systems
consisting of odd functions of variables and symplectic lattices of density
cannot constitute a Gabor frame. In the 1--dimensional, separable case,
this is a special case of a result proved by Lyubarskii and Nes, however, we
use a different approach in this work exploiting the algebraic relation between
the ambiguity function and the Wigner distribution as well as their relation
given by the (symplectic) Fourier transform. Also, we do not need the
assumption that the lattice is separable and, hence, new restrictions are added
to the full frame set of odd functions.Comment: accepted: Bulletin of the Australian Mathematical Society; 12 pages;
Version 3 makes use of symmetric time-frequency shifts. In this case the
appearing phase factors are easier to handle. Also, the main result is
extended to higher dimensions. [In Version 2 a mistake in the assumptions was
corrected. The windows should be chosen from Feichtinger's algebra rather
than from the Hilbert space L2.
Spectral asymptotics of periodic elliptic operators
We demonstrate that the structure of complex second-order strongly elliptic
operators on with coefficients invariant under translation by
can be analyzed through decomposition in terms of versions ,
, of with -periodic boundary conditions acting on
where . If the semigroup generated by
has a H\"older continuous integral kernel satisfying Gaussian bounds then the
semigroups generated by the have kernels with similar properties
and extends to a function on which is
analytic with respect to the trace norm. The sequence of semigroups
obtained by rescaling the coefficients of by converges in
trace norm to the semigroup generated by the homogenization
of . These convergence properties allow asymptotic analysis of
the spectrum of .Comment: 27 pages, LaTeX article styl
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