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

We study time-frequency localization operators of the form A a φ 1 , φ 2 A_a^{\varphi _1\!,\varphi _2} , where a a is the symbol of the operator and φ 1 , φ 2 \varphi _1 , \varphi _2 are the analysis and synthesis windows, respectively. It is shown in an earlier paper by the authors that a sufficient condition for A a φ 1 , φ 2 ∈ S p ( L 2 ( R d ) ) A_a^{\varphi _1,\varphi _2}\in S_p(L^2(\mathbb {R}^d)) , the Schatten class of order p p , is that a a belongs to the modulation space M p , ∞ ( R 2 d ) M^{p,\infty }(\mathbb {R}^{2d}) and the window functions to the modulation space M 1 M^1 . Here we prove a partial converse: if A a φ 1 , φ 2 ∈ S p ( L 2 ( R d ) ) A_a^{\varphi _1,\varphi _2}\in S_p(L^2(\mathbb {R}^d)) for every pair of window functions φ 1 , φ 2 ∈ S ( R 2 d ) \varphi _1,\varphi _2\in \mathcal {S}(\mathbb {R}^{2d}) with a uniform norm estimate, then the corresponding symbol a a must belong to the modulation space M p , ∞ ( R 2 d ) M^{p,\infty }(\mathbb {R}^{2d}) . In this sense, modulation spaces are optimal for the study of localization operators. The main ingredients in our proofs are frame theory and Gabor frames. For p = ∞ p=\infty and p = 2 p=2 , we recapture earlier results, which were obtained by different methods.

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

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.

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.

This is a survey about the theory of Gabor frames. We review the structural results about Gabor frames over a lattice and then discuss the few known results about the fine structure of Gabor frames. We add a new result about the relation between properties of the window and properties of the frame set and conclude with a vision of how a more complete theory of the fine structure might look like.

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.

A technique for producing signals whose energy is concentrated in a given region of the time-frequency plane is examined. The degree to which a given signal is concentrated in a region is measured by integrating its time-frequency distribution over the region. This method has been used for time-varying filtering. Localization operators based on the Wigner distribution and spectrogram are studied. Estimates for the eigenvalue decay and the smoothness and decay of the eigenfunctions are presented. >

Vector‐valued frames were first introduced under the name of superframes by Balan in the context of signal multiplexing and by Han and Larson from the mathematical aspect. Since then, the wavelet and Gabor frames in have interested many mathematicians. The space models vector‐valued causal signal spaces because of the time variable being nonnegative. But it admits no nontrivial shift‐invariant system and thus no wavelet or Gabor frame since is not a group by addition (not as ). Observing that is a group by multiplication, we, in this paper, introduce a class of multiplication‐based dilation‐and‐modulation ( ) systems, and investigate the theory of frames in . Since is not closed under the Fourier transform, the Fourier transform does not fit . We introduce the notion of Θa transform in , and using Θa‐transform matrix method, we characterize frames, Riesz bases, and dual frames in and obtain an explicit expression of duals for an arbitrary given frame. An example theorem is also presented.

The main issue in Chapter 11 was to state necessary and/or sufficient conditions for a Gabor system \(\{E_{mb}T_{na}g\}_{m,n\in \mathbb{Z}}\) in \(L^{2}(\mathbb{R})\) to form a frame. We will now take the next step and consider Gabor frames that are convenient to apply in practice. From the general frame theory in Chapter 5 and Chapter 6 we know that frames are particularly useful when the frame decomposition takes a simple form, which is the case if either the frame is tight or we have access to a convenient dual frame.

This paper addresses multi-window Gabor frames with ratio- nal time-frequency product. Such issue was considered by Zibulski and Zeevi (Appl. Comput. Harmonic Anal. 4 (1997), 188-221) in terms of Zak transform matrix (so-called Zibuski-Zeevi matrix), and by many oth- ers. In this paper, we introduce of a new Zak transform matrix. It is different from Zibulski-Zeevi matrix, but more direct and convenient for our purpose. Using such Zak transform matrix we characterize rational time-frequency multi-window Gabor frames (Riesz bases and orthonor- mal bases), and Gabor duals for a Gabor frame. Some examples are also provided, which show that our Zak transform matrix method is efficient.

The Weyl correspondence is a convenient and practical way to define a broad class of time-frequency localization operators. Given a region /spl Omega/ in the time-frequency (or phase) plane R/sup 2/, and a distribution P(/spl xi/, x) in the Cohen's (1989) class, the Weyl correspondence defines an operator L/sub /spl Omega///sup P/ on L/sup 2/(R) which essentially localizes the signal in the region /spl Omega/ according to the distribution P. Different choices of P allow different interpretations of phase-space localization. Empirically, such localization operators have the following singular value structure: the singular values range between 0 and 1, and when arranged in decreasing order, a finite number of them are close to 1, while the rest fall toward 0 asymptotically. We present a result on the asymptotic structure of the singular values of such phase-space localization operators. >

We consider the spaces of ultradifferentiable functions $$\mathcal {S}_\omega $$ as introduced by Bjorck (and its dual $$\mathcal {S}'_\omega $$ ) and we use time-frequency analysis to define a suitable wave front set in this setting and obtain several applications: global regularity properties of pseudodifferential operators of infinite order and the micro-pseudolocal behaviour of partial differential operators with polynomial coefficients and of localization operators with symbols of exponential growth. Moreover, we prove that the new wave front set, defined in terms of the Gabor transform, can be described using only Gabor frames. Finally, some examples show the convenience of the use of weight functions to describe more precisely the global regularity of (ultra)distributions.

Analogs of classical results on accumulated spectrograms, the sum of spectrograms of eigenfunctions of localization operators, are established for Gabor multipliers. We show that the lattice ℓ1 distance between the accumulated spectrogram and the indicator function of the Gabor multiplier mask is bounded by the number of lattice points near the boundary of the mask and that this bound is sharp in general. The methods developed for the proofs are also used to show that the Weyl-Heisenberg ensemble restricted to a lattice is hyperuniform when the Gabor frame is tight.

Local and semilocal vertex operator algebras

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.