200 research outputs found
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
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
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
Measurement of time--varying Multiple--Input Multiple--Output Channels
We derive a criterion on the measurability / identifiability of
Multiple--Input Multiple--Output (MIMO) channels based on the size of the
so-called spreading support of its subchannels. Novel MIMO transmission
techniques provide high-capacity communication channels in time-varying
environments and exact knowledge of the transmission channel operator is of key
importance when trying to transmit information at a rate close to channel
capacity
The Reconstruction Problem and Weak Quantum Values
Quantum Mechanical weak values are an interference effect measured by the
cross-Wigner transform W({\phi},{\psi}) of the post-and preselected states,
leading to a complex quasi-distribution {\rho}_{{\phi},{\psi}}(x,p) on phase
space. We show that the knowledge of {\rho}_{{\phi},{\psi}}(z) and of one of
the two functions {\phi},{\psi} unambiguously determines the other, thus
generalizing a recent reconstruction result of Lundeen and his collaborators.Comment: To appear in J.Phys.: Math. Theo
Conormal distributions in the Shubin calculus of pseudodifferential operators
We characterize the Schwartz kernels of pseudodifferential operators of
Shubin type by means of an FBI transform. Based on this we introduce as a
generalization a new class of tempered distributions called Shubin conormal
distributions. We study their transformation behavior, normal forms and
microlocal properties.Comment: 23 page
The Gabor wave front set of compactly supported distributions
We show that the Gabor wave front set of a compactly supported distribution
equals zero times the projection on the second variable of the classical wave
front set
Compactly supported wavelets and representations of the Cuntz relations, II
We show that compactly supported wavelets in L^2(R) of scale N may be
effectively parameterized with a finite set of spin vectors in C^N, and
conversely that every set of spin vectors corresponds to a wavelet. The
characterization is given in terms of irreducible representations of
orthogonality relations defined from multiresolution wavelet filters.Comment: 10 or 11 pages, SPIE Technical Conference, Wavelet Applications in
Signal and Image Processing VII
Distances sets that are a shift of the integers and Fourier basis for planar convex sets
The aim of this paper is to prove that if a planar set has a difference
set satisfying for suitable than
has at most 3 elements. This result is motivated by the conjecture that the
disk has not more than 3 orthogonal exponentials. Further, we prove that if
is a set of exponentials mutually orthogonal with respect to any symmetric
convex set in the plane with a smooth boundary and everywhere non-vanishing
curvature, then # (A \cap {[-q,q]}^2) \leq C(K) q where is a constant
depending only on . This extends and clarifies in the plane the result of
Iosevich and Rudnev. As a corollary, we obtain the result from \cite{IKP01} and
\cite{IKT01} that if is a centrally symmetric convex body with a smooth
boundary and non-vanishing curvature, then does not possess an
orthogonal basis of exponentials
Frame expansions for Gabor multipliers
AbstractDiscrete Gabor multipliers are composed of rank one operators. We shall prove, in the case of rank one projection operators, that the generating operators for such multipliers are either Riesz bases (exact frames) or not frames for their closed linear spans. The same dichotomy conclusion is valid for general rank one operators under mild and natural conditions. This is relevant since discrete Gabor multipliers have an emerging role in communications, radar, and waveform design, where redundant frame decompositions are increasingly applicable
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