Subexponential decay and regularity estimates for eigenfunctions of 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.

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.

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. >

In the last twenty years modulation spaces, introduced by H. G. Feichtinger in 1983, have been successfully addressed to the study of signal analysis, PDE's, pseudodifferential operators, quantum mechanics, by hundreds of contributions. In 2011 M. de Gosson showed that the time-frequency representation Short-time Fourier Transform (STFT), which is the tool to define modulation spaces, can be replaced by the Wigner distribution. This idea was further generalized to τ-Wigner representations in [11].In this paper time-frequency representations are viewed as images of symplectic matrices via metaplectic operators. This new perspective highlights that the protagonists of time-frequency analysis are metaplectic operators and symplectic matrices A∈Sp(2d,R). We find conditions on A for which the related symplectic time-frequency representation WA can replace the STFT and give equivalent norms for weighted modulation spaces. In particular, we study the case of covariant matrices A, i.e., their corresponding WA are members of the Cohen class.Finally, we show that symplectic time-frequency representations WA can be efficiently employed in the study of Schrödinger equations. In fact, modulation spaces and WA representations are the frame for a new definition of wave front set, providing a sharp result for propagation of micro-singularities in the case of the quadratic Hamiltonians. This new approach may have further applications in quantum mechanics and PDE's.

We now come to the mathematical foundation of quantum mechanics on phase space. In the two previous chapters, motivation for and some of the desired mathematical structure of quantum mechanics on phase space has emerged. In this chapter, we give a rigorous mathematical analysis of the origin, the variety, and the structure of quantum mechanics on phase space. To review, the discovered components of the structure of quantum mechanics on phase space are: (i) A symmetry group; (ii) A classical phase space, Γ; (iii) Representations of the symmetry group by unitary operators acting on the Hilbert space L2(Γ); (iv) Localization operators (P.V.M.’s) on these representations; (v) Harmonic decompositions of these representations into irreducible sub-representations; (vi) Intertwining operators between these irreducible sub-representations and the standard irreducible representations [as commonly used in physics and obtained by the Mackey-Frobenius method of induced representations.] The intertwined images of the (P.V.M.) localization operators become P.O.V.M.’s. (vii) Informational completeness criteria for these P.O.V.M.’s.

Recently, intelligent reflecting surfaces (IRSs) have emerged as potential candidates for overcoming the line-of-sight issue in 5 G/6 G wireless communication. These IRSs can manipulate the direction of reflected beams, enabling efficient beam steering to enhance the performance of wireless communication. Each unit cell (or unit structure) of an IRS commonly consists of electrical elements for phase modulation. However, by employing phase modulation alone, an IRS can steer the reflected electromagnetic waves toward only discrete and specific angles, leaving a wide range of out-of-beam areas. In this work, an IRS that uses both phase modulation and space modulation is presented to improve the beam resolution and continuously cover out-of-beam areas that phase modulation alone cannot address. A positive-intrinsic-negative diode is mounted on a unit cell for phase modulation, and a 4D-printed reconfigured structure is fabricated to demonstrate space modulation. The beam-steering function is achieved by alternating the states of the diodes in the same columns, while the beam resolution is improved by controlling the gaps between the columns. The functions are first theoretically and numerically analyzed and then experimentally verified, demonstrating that additional angles of −46°/+50°, −22°/+14°, and −16°/+12° are achieved with space modulation and −60°/+62°, −30°/+22°, and ±16° are achieved by phase modulation alone. The proposed IRS offers the possibility of functional integration in a variety of indoor applications within the wireless communication field.

We introduce new classes of modulation spaces over phase space. By means of the Kohn-Nirenberg correspondence, these spaces induce norms on pseudo-differential operators that bound their operator norms on Lp–spaces, Sobolev spaces, and modulation spaces.

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

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. >

<abstract><p>In this paper, we consider the Cauchy problem for a family of evolution-parabolic coupled systems, which are related to the classical thermoelastic plate equations containing non-local operators. By using diagonalization procedure and WKB analysis, we derive representation of solutions in the phase space. Then, sharp decay properties in a framework of $ L^p-L^q $ are investigated via these representations. Particularly, some thresholds for the regularity-loss type decay properties are found.</p></abstract>

Semi-Lagrangian schemes for the Vlasov equation on an unstructured mesh of phase space

We investigate some properties for Hilbert -modular frames. In the present paper, we also investigate some conditions for invertibility of multipliers in Hilbert -module spaces. We show that by appropriate conditions for symbols, the inverse of an invertible frame multiplier can be represented as a frame multiplier. Dual frames play important role in this representation. We prove and generalize some of the results obtained for frames in Hilbert spaces to Hilbert -modules.

We are interested in studying models of thermoelasticity from a unified point of view, i.e. we are studying Cauchy problems for linear second-order systems of the form Ut + A0U + A1Ux − A2Uxx = 0 for d-dimensional unknowns U = U(t, x) with (t, x) ∈ R>0 × R and Ai being real and constant d× d matrices. Using a diagonalization procedure in phase space, we are able to obtain (under certain conditions) solution representations and to apply these to the derivation of well-posedness results, L–L decay estimates and diffusion phenomena.

We use sharp convolution estimates for weighted Lebesgue and modulation spaces to obtain an extension of the celebrated Cordero-Grochenig theorems on boundedness and Schatten–von Neumann properties of localization operators on modulation spaces. We also give a new proof of the Weyl connection based on the kernel theorem for Gelfand–Shilov spaces.

We study the fractal uncertainty principle in the joint time-frequency representation, and we prove a version for the Short-Time Fourier transform with Gaussian window on the modulation spaces. This can equivalently be formulated in terms of projection operators on the Bargmann-Fock spaces of entire functions. Specifically for signals in L 2 ( R d ) , we obtain norm estimates of Daubechies' time-frequency localization operator localizing on porous sets. The proof is based on the maximal Nyquist density of such sets, which we also use to derive explicit upper bound asymptotes for the multidimensional Cantor iterates, in particular. Finally, we translate the fractal uncertainty principle to discrete Gaussian Gabor multipliers.