Necessary conditions for Schatten class localization operators

Time-frequency partitions and characterizations of modulation spaces with localization operators

We consider time-frequency localization operators A_a^{varphi _1,varphi _2} with symbols a in the wide weighted modulation space M^infty _{w}({mathbb {R}^{2d}}), and windows varphi _1, varphi _2 in the Gelfand–Shilov space mathcal {S}^{left( 1right) }(mathbb {R}^d). If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of A_a^{varphi _1,varphi _2} have appropriate subexponential decay in phase space, i.e. that they belong to the Gelfand–Shilov space mathcal {S}^{(gamma )} (mathbb {R^{d}}) , where the parameter gamma ge 1 is related to the growth of the considered weight. An important role is played by tau -pseudodifferential operators Op_{tau } (sigma ). In that direction we show convenient continuity properties of Op_{tau } (sigma )when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of Op_{tau } (sigma ) when the symbol sigma belongs to a modulation space with appropriately chosen weight functions. As an auxiliary result we also prove new convolution relations for (quasi-)Banach weighted modulation spaces.

Here, we analyze sparse analysis windows for complex-valued and real-valued discrete Gabor transforms. A number of theories presented in this study indicate that the existence and uniqueness of a sparse analysis window for a given synthesis window depend to a great extent on the sparsity of the synthesis window. Specifically, the upper and lower bounds on the analysis window sparsity are obtained for synthesis windows of different sparsities and analyzed in detail. Particularly, if the sparsity of a given synthesis window with connected support is equal to the discrete Gabor transform time-shift parameter, the sparsest analysis window is unique. In addition, we propose an algorithm that searches for a sparse analysis window. Compared with existing algorithms, a sparse analysis window with fewer nonzero elements and/or smaller reconstruction errors can be obtained with the proposed algorithm. A sequence of experimental results reveals its superior effectiveness. In addition, a better Gabor representation with high concentration and time–frequency resolution is obtained with the sparse analysis window.

The Quantization of Edge Symbols.- On Rays of Minimal Growth for Elliptic Cone Operators.- Symbolic Calculus of Pseudo-differential Operators and Curvature of Manifolds.- Weyl Transforms, Heat Kernels, Green Functions and Riemann Zeta Functions on Compact Lie Groups.- On the Fourier Analysis of Operators on the Torus.- Wave Kernels of the Twisted Laplacian.- Super-exponential Decay of Solutions to Differential Equations in ?d.- Gevrey Local Solvability for Degenerate Parabolic Operators of Higher Order.- A New Aspect of the L p-extension Problem for Inhomogeneous Differential Equations.- Continuity in Quasi-homogeneous Sobolev Spaces for Pseudo-differential Operators with Besov Symbols.- Continuity and Schatten Properties for Pseudo-differential Operators on Modulation Spaces.- Algebras of Pseudo-differential Operators with Discontinuous Symbols.- A Class of Quadratic Time-frequency Representations Based on the Short-time Fourier Transform.- A Characterization of Stockwell Spectra.- Exact and Numerical Inversion of Pseudo-differential Operators and Applications to Signal Processing.- On the Product of Localization Operators.- Gelfand-Shilov Spaces, Pseudo-differential Operators and Localization Operators.- Continuity and Schatten Properties for Toeplitz Operators on Modulation Spaces.- Microlocalization within Some Classes of Fourier Hyperfunctions.

Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces

An efficient algorithm to reduce the noise from the Nuclear Magnetic Resonance Free Induction Decay (NMR FID) signals is presented, in this paper, via the oversampled real-valued discrete Gabor transform using the Gaussian synthesis window. An NMR FID signal in the Gabor transform domain (i.e., a joint time-frequency domain) is concentrated in a few number of Gabor transform coefficients while the noise is fairly distributed among all the coefficients. Therefore, the NMR FID signal can be significantly enhanced by performing a thresholding technique on the coefficients in the transform domain. Theoretical and simulation experimental analyses in this paper show that the oversampled Gabor transform using the Gaussian synthesis window is more suitable for the NMR FID signal enhancement than the critically-sampled one using the exponential synthesis window, because both the Gaussian synthesis window and its corresponding analysis window in the oversampling case can have better localization in the frequency domain than the exponential synthesis window and its corresponding analysis window in the critically-sampling case. Moreover, to speed up the transform, instead of the commonly-used complex-valued discrete Gabor transform, the real-valued discrete Gabor transform presented in our previous work is adopted in the proposed algorithm.

Well-posedness of the modified dispersion-generalized Benjamin-Ono equations in modulation spaces

We deduce factorization properties for Wiener amalgam spaces WLp,q, an extended family of modulation spaces M(ω,ℬ), and for Schatten symbols spw in pseudo-differential calculus under e.g. convolutions, twisted convolutions and symbolic products. Here M(ω,ℬ) can be any quasi-Banach Orlicz modulation space. For example we show that WL1,r∗WLp,q=WLp,q and WL1,r#spw=spw when r∈(0,1], r≤p,q<∞. In particular we improve Rudin’s identity L1∗L1=L1.

We study when a multipartite non--local unitary operation can deterministically or probabilistically simulate another one when local operations of a certain kind ---in some cases including also classical communication--- are allowed. In the case of probabilistic simulation and allowing for arbitrary local operations, we provide necessary and sufficient conditions for the simulation to be possible. Deterministic and probabilistic interconversion under certain kinds of local operations are used to define equivalence relations between gates. In the probabilistic, bipartite case this induces a finite number of classes. In multiqubit systems, however, two unitary operations typically cannot simulate each other with non-zero probability of success. We also show which kind of entanglement can be created by a given non--local unitary operation and generalize our results to arbitrary operators.

We study nonuniform sampling in shift-invariant spaces and the construction of Gabor frames with respect to the class of totally positive functions whose Fourier transform factors as hat{g}(xi )= prod _{j=1}^n (1+2pi idelta _jxi )^{-1} , e^{-c xi ^2} for delta _1,ldots ,delta _nin mathbb {R}, c >0 (in which case g is called totally positive of Gaussian type). In analogy to Beurling’s sampling theorem for the Paley–Wiener space of entire functions, we prove that every separated set with lower Beurling density >1 is a sampling set for the shift-invariant space generated by such a g. In view of the known necessary density conditions, this result is optimal and validates the heuristic reasonings in the engineering literature. Using a subtle connection between sampling in shift-invariant spaces and the theory of Gabor frames, we show that the set of phase-space shifts of g with respect to a rectangular lattice alpha mathbb {Z}times beta mathbb {Z} forms a frame, if and only if alpha beta <1. This solves an open problem going back to Daubechies in 1990 for the class of totally positive functions of Gaussian type. The proof strategy involves the connection between sampling in shift-invariant spaces and Gabor frames, a new characterization of sampling sets “without inequalities” in the style of Beurling, new properties of totally positive functions, and the interplay between zero sets of functions in a shift-invariant space and functions in the Bargmann–Fock space.

G?vruta studied atomic systems in terms of frames for range of operators (that is, for subspaces), namely ?-frames, where the lower frame condition is controlled by the Hilbert-adjoint of a bounded linear operator?. For a locally compact abelian groupGand a positive integer n, westudy frames of matrix-valued Gabor systems in the matrix-valued Lebesgue space L2(G,Cn?n) , where a bounded linear operator ? on L2(G,Cn?n) controls not only lower but also the upper frame condition. We term such frames matrix-valued (?,?*)-Gabor frames. Firstly, we discuss frame preserving mapping in terms of hyponormal operators. Secondly, we give necessary and sufficient conditions for the existence of matrix-valued (?,?*)- Gabor frames in terms of hyponormal operators. It is shown that if ? is adjointable hyponormal operator, then L2(G,Cn?n) admits a ?-tight (?,?*)-Gabor frame for every positive real number ?. A characterization of matrix-valued (?,?*)-Gabor frames is given. Finally, we show that matrix-valued (?,?*)-Gabor frames are stable under small perturbation of window functions. Several examples are given to support our study.

Motivated by the human auditory system, a new signal transform is presented which models the way humans hear. Cochlear processing acts like a constant bandwidth bank of filters in the low frequency range but is of proportional bandwidth at higher frequencies. This new transform, which we call the composite wavelet transform, is better able to model this process than standard signal processing techniques such as the short‐time Fourier transform (STFT) and the continuous wavelet transform (CWT). The composite wavelet transform in fact provides a signal analysis tool that is able to examine signals with competing signal structures whereas the STFT and the CWT do not. In order to insure stable signal recovery, the theory of frames is examined. An overview of the theory of frames will be given and wavelet and Gabor frames will be discussed. Recent work on frames for the composite wavelet transform will also be presented. Finally, some applications of frames, especially to speech signal processing will be considered.

To efficiently analyze the dynamic time-frequency contents of signals that contain a wide range of time and frequency components, a new real-valued discrete Gabor transform with multi-windows M-RDGT based on the bi-orthogonal analysis approach is presented in this paper, which permits a computationally faster implementation as an alternate formulation of the complex-valued discrete Gabor transform with multi-windows M-CDGT. The completeness condition of the M-RDGT is proved to be equivalent to its bi-orthogonality constraint between analysis windows and synthesis windows. The M-RDGT is defined by replacing the complex-valued Gabor basis functions of the M-CDGT with real-valued Gabor basis functions. The real-valued Gabor basis functions of the M-RDGT contain the Hartley's cas function which allows the M-RDGT to utilize the fast discrete Hartley transform algorithms for fast computation. In addition, the M-RDGT has a simple relationship with the M-CDGT such that the M-CDGT coefficients can be directly computed from the M-RDGT coefficients. Therefore, the M-RDGT also offers a faster and more efficient method to compute the M-CDGT.

Based on simulations of denoising of Gabor coefficient spectrum with Gaussian dual windows, threshold filter denoising and morphological Filter denoising methods have been implemented. With conventional Gabor order tracking algorithm, Gaussian window applied as analysis window and synthesis window, a trail-and-error threshold filter denoising method reconstructed original signal without noise, the choose of value of threshold filter is a waste of time. However, morphological Filter denoising is put forward as a alternative vision solutions, flood fill algorithm looks for all nodes in the array that are connected to the start node by a path of the target color and keep this area unchanged. Mask operation sets unselected region to zero, simulations indicate that should be rapid and effective.

Local operators are characterized mathematically by means of projection operators on the Banach space of bounded operators. The idea of microlocality, as opposed to macrolocality, is implemented into the theory so as to enable us to define operations that are strictly local. Necessary and sufficient conditions are investigated in order that the interaction of a local measurement instrument with a local quantum field is such a strictly local (or microlocal) operation. Application of the theory to quantum electrodynamics reveals that this theory violates microlocality as defined here. Implications which our theory may have on the issue of quantum nonlocality as studied in relation to the Bell inequalities are discussed.