383 research outputs found
Landau's necessary density conditions for LCA groups
H. Landau's necessary density conditions for sampling and interpolation may
be viewed as a general principle resting on a basic fact of Fourier analysis:
The complex exponentials ( in ) constitute an
orthogonal basis for . The present paper extends Landau's
conditions to the setting of locally compact abelian (LCA) groups, relying in
an analogous way on the basics of Fourier analysis. The technicalities--in
either case of an operator theoretic nature--are however quite different. We
will base our proofs on the comparison principle of J. Ramanathan and T.
Steger
Frame Constants of Gabor Frames near the Critical Density
We consider Gabor frames generated by a Gaussian function and describe the
behavior of the frame constants as the density of the lattice approaches the
critical value
A Guide to Localized Frames and Applications to Galerkin-like Representations of Operators
This chapter offers a detailed survey on intrinsically localized frames and
the corresponding matrix representation of operators. We re-investigate the
properties of localized frames and the associated Banach spaces in full detail.
We investigate the representation of operators using localized frames in a
Galerkin-type scheme. We show how the boundedness and the invertibility of
matrices and operators are linked and give some sufficient and necessary
conditions for the boundedness of operators between the associated Banach
spaces.Comment: 32 page
On the Usefulness of Modulation Spaces in Deformation Quantization
We discuss the relevance to deformation quantization of Feichtinger's
modulation spaces, especially of the weighted Sjoestrand classes. These
function spaces are good classes of symbols of pseudo-differential operators
(observables). They have a widespread use in time-frequency analysis and
related topics, but are not very well-known in physics. It turns out that they
are particularly well adapted to the study of the Moyal star-product and of the
star-exponential.Comment: Submitte
Approximation of Fourier Integral Operators by Gabor multipliers
A general principle says that the matrix of a Fourier integral operator with
respect to wave packets is concentrated near the curve of propagation. We prove
a precise version of this principle for Fourier integral operators with a
smooth phase and a symbol in the Sjoestrand class and use Gabor frames as wave
packets. The almost diagonalization of such Fourier integral operators suggests
a specific approximation by (a sum of) elementary operators, namely modified
Gabor multipliers. We derive error estimates for such approximations. The
methods are taken from time-frequency analysis.Comment: 22. page
An Inverse Problem for Localization Operators
A classical result of time-frequency analysis, obtained by I. Daubechies in
1988, states that the eigenfunctions of a time-frequency localization operator
with circular localization domain and Gaussian analysis window are the Hermite
functions. In this contribution, a converse of Daubechies' theorem is proved.
More precisely, it is shown that, for simply connected localization domains, if
one of the eigenfunctions of a time-frequency localization operator with
Gaussian window is a Hermite function, then its localization domain is a disc.
The general problem of obtaining, from some knowledge of its eigenfunctions,
information about the symbol of a time-frequency localization operator, is
denoted as the inverse problem, and the problem studied by Daubechies as the
direct problem of time-frequency analysis. Here, we also solve the
corresponding problem for wavelet localization, providing the inverse problem
analogue of the direct problem studied by Daubechies and Paul.Comment: 18 pages, 1 figur
Balian-Low Theorems in Several Variables
Recently, Nitzan and Olsen showed that Balian-Low theorems (BLTs) hold for
discrete Gabor systems defined on . Here we extend these results
to a multivariable setting. Additionally, we show a variety of applications of
the Quantitative BLT, proving in particular nonsymmetric BLTs in both the
discrete and continuous setting for functions with more than one argument.
Finally, in direct analogy of the continuous setting, we show the Quantitative
Finite BLT implies the Finite BLT.Comment: To appear in Approximation Theory 16 conference proceedings volum
A characterization of Schauder frames which are near-Schauder bases
A basic problem of interest in connection with the study of Schauder frames
in Banach spaces is that of characterizing those Schauder frames which can
essentially be regarded as Schauder bases. In this paper, we give a solution to
this problem using the notion of the minimal-associated sequence spaces and the
minimal-associated reconstruction operators for Schauder frames. We prove that
a Schauder frame is a near-Schauder basis if and only if the kernel of the
minimal-associated reconstruction operator contains no copy of . In
particular, a Schauder frame of a Banach space with no copy of is a
near-Schauder basis if and only if the minimal-associated sequence space
contains no copy of . In these cases, the minimal-associated
reconstruction operator has a finite dimensional kernel and the dimension of
the kernel is exactly the excess of the near-Schauder basis. Using these
results, we make related applications on Besselian frames and near-Riesz bases.Comment: 12 page
An Entropy Based Method for Local Time-Adaptation of the Spectrogram
We propose a method for automatic local time-adaptation of the spectrogram of
audio signals: it is based on the decomposition of a signal within a Gabor
multi-frame through the STFT operator. The sparsity of the analysis in every
individual frame of the multi-frame is evaluated through the R\'enyi entropy
measures: the best local resolution is determined minimizing the entropy
values. The overall spectrogram of the signal we obtain thus provides local
optimal resolution adaptively evolving over time. We give examples of the
performance of our algorithm with an instrumental sound and a synthetic one,
showing the improvement in spectrogram displaying obtained with an automatic
adaptation of the resolution. The analysis operator is invertible, thus leading
to a perfect reconstruction of the original signal through the analysis
coefficients
Linear perturbations of the Wigner transform and the Weyl quantization
We study a class of quadratic time-frequency representations that, roughly
speaking, are obtained by linear perturbations of the Wigner transform. They
satisfy Moyal's formula by default and share many other properties with the
Wigner transform, but in general they do not belong to Cohen's class. We
provide a characterization of the intersection of the two classes. To any such
time-frequency representation, we associate a pseudodifferential calculus. We
investigate the related quantization procedure, study the properties of the
pseudodifferential operators, and compare the formalism with that of the Weyl
calculus.Comment: 38 pages. Contributed chapter for volume on the occasion of Luigi
Rodino's 70th birthda
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