For $m\in\mathbb{R}$ we introduce the symbol classes $S^m$, $m\in\mathbb{R}$, consisting of smooth functions $σ$ on $\mathbb{R}^{2d}$ such that $|\partial^ασ(z)|\leq C_α(1+|z|^2)^{m/2}$, $z\in\mathbb{R}^{2d}$, and we show that can be characterized by an intersection of different types of modulation spaces. In the case $m=0$ we recapture the Hörmander class $S^0_{0,0}$ that can be obtained by intersection of suitable Besov spaces as well. Such spaces contain the Shubin classes $Γ^m_ρ$, $0<ρ\leq1$, and can be viewed as their limit case $ρ=0$. We exhibit almost diagonalization properties for the Gabor matrix of $τ$-pseudodifferential operators with symbols in such classes, extending the characterization proved by Gröchenig and Rzeszotnik. Finally, we compute the Gabor matrix of a Born-Jordan operator, which allows to prove new boundedness results for such operators.
We investigate a new representation of general operators by means of sums of shifted Gabor multipliers. These representations arise by studying the matrix of an operator with respect to a Gabor frame. Each shifted Gabor multiplier corresponds to a side-diagonal of this matrix. This representation is especially useful for operators whose associated matrix possesses some off-diagonal decay. In this case one can completely characterize the symbol class of the operator by the size of the symbols of the Gabor multipliers. As an application we derive approximation theorems for pseudodifferential ope…
Motivated by the problem of channel estimation in wireless communications, we derive a reconstruction formula for pseudodifferential operators with a bandlimited symbol. This reconstruction formula uses the diagonal entries of the matrix of the pseudodifferential operator with respect to a Gabor system. In addition, we prove several other uniqueness theorems that shed light on the relation between a pseudodifferential operators and its matrix with respect to a Gabor system.
We study one-parameter families of pseudodifferential operators whose Weyl symbols are obtained by dilation and a smooth deformation of a symbol in a weighted Sjöstrand class. We show that their spectral edges are Lipschitz continuous functions of the dilation or deformation parameter. Suitably local estimates hold also for the edges of every spectral gap. These statements extend Bellissard's seminal results on the Lipschitz continuity of spectral edges for families of operators with periodic symbols to a large class of symbols with only mild regularity assumptions. The abstract results are us…
The geometry of fundamental domains of lattices was used by Han and Wang to construct multivariate Gabor frames for separable lattices. We build upon their results to obtain Gabor frames with smooth and compactly supported window functions. For this purpose we study pairs of lattices which have equal density and allow for a common compact and star-shaped fundamental domain. The results are then extended to a larger class of lattices via symplectic equivalence.
We consider Scrödinger equations with real-valued smooth Hamiltonians, and non-smooth bounded pseudo-differential potentials, whose symbols may be not even differentiable. The well-posedness of the Cauchy problem is proved in the frame of the modulation spaces, and results of micro-local propagation of singularities are given in terms of Gabor wave front sets.
We characterize the entire functions $P$ of $d$ variables, $d\ge 2,$ for which the $\mzd$-translates of $Pχ_{[0,N]^d}$ satisfy the partition of unity for some $N\in \mn.$ In contrast to the one-dimensional case, these entire functions are not necessarily periodic. In the case where $P$ is a trigonometric polynomial, we characterize the maximal smoothness of $Pχ_{[0,N]^d},$ as well as the function that achieves it. A number of especially attractive constructions are achieved, e.g., of trigonometric polynomials leading to any desired (finite) regularity for a fixed support size. As an applicatio…
We perform a time-frequency analysis of Fourier multipliers and, more generally, pseudodifferential operators with symbols of Gevrey, analytic and ultra-analytic regularity. As an application we show that Gabor frames, which provide optimally sparse decompositions for Schroedinger-type propagators, reveal to be an even more efficient tool for representing solutions to a wide class of evolution operators with constant coefficients, including weakly hyperbolic and parabolic-type operators. Besides the class of operators, the main novelty of the paper is the proof of super-exponential (as opposit…
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.
An explicit expression for the Kohn-Nirenberg symbol of a Weyl- Heisenberg frame operator on $L^2(\mathbb{R})$ is obtained directly from the Gabor atom coming from new classes of window functions. This new approach, using only elementary Fourier analysis, is independent of the theory of distributions and works strictly inside $L^2(\mathbb{R})$. Kohn-Nirenberg operators are introduced and are shown to be Weyl-Heisenberg frame operators in suitable cases.
Let $Φ$ and $Ψ$ be frames for $\cal H$ and let $M_{m,Φ,Ψ}$ be a frame multiplier with the symbol $m$. In this paper, we restrict our investigation to show that the operator properties of $M_{m,Φ,Ψ}$ are stable under the perturbations of $Φ$, $Ψ$ and $m$. Also, special attention is devoted to the study of invertible frame multipliers. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Gabor multipliers which have been studied mostly by Hans Georg Feichtinger and his coauthors in recent years.
We consider a class of linear Schroedinger equations in R^d, with analytic symbols. We prove a global-in-time integral representation for the corresponding propagator as a generalized Gabor multiplier with a window analytic and decaying exponentially at infinity, which is transported by the Hamiltonian flow. We then provide three applications of the above result: the exponential sparsity in phase space of the corresponding propagator with respect to Gabor wave packets, a wave packet characterization of Fourier integral operators with analytic phases and symbols, and the propagation of analytic…
We present a unified approach to study properties of Toeplitz localization operators based on the Calderón and Gabor reproducing formula. We show that these operators with functional symbols on a plane domain may be viewed as certain pseudodifferential operators (with symbols on a line, or certain compound symbols).
We consider the problem of reconstructing a signal $f$ from its spectrogram, i.e., the magnitudes $|V_φf|$ of its Gabor transform $$V_φf (x,y):=\int_{\mathbb{R}}f(t)e^{-π(t-x)^2}e^{-2πıy t}dt, \quad x,y\in \mathbb{R}.$$ Such problems occur in a wide range of applications, from optical imaging of nanoscale structures to audio processing and classification. While it is well-known that the solution of the above Gabor phase retrieval problem is unique up to natural identifications, the stability of the reconstruction has remained wide open. The present paper discovers a deep and surprising connect…
We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the Kohn-Nirenberg calculus we introduce a version of Sjoestrand's class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since "hard analysis" techniques are not available on locally compact abelian groups, a new time-frequency approach is used with the emphasis on modulation spaces, Gabor frames, and Banach algebras of matrices. Sjoestrand's original results are thus understood as a phenomenon of abstract harmonic analysis rather …
In this article, we study properties of multilinear Fourier integral operators on weighted modulation spaces. In particular, using the theory of Gabor frames, we study boundedness of multilinear Fourier integral operators on products of weighted modulation spaces. Further, we investigate the periodic multilinear Fourier integral operator. Finally, we study continuity of bilinear pseudo-differential operators on modulation spaces for certain symbol classes, namely $\textbf{SG}$-class.
In this review we focus on the almost diagonalization of pseudodifferential operators and highlight the advantages that time-frequency techniques provide here. In particular, we retrace the steps of an insightful paper by Gröchenig, who succeeded in characterizing a class of symbols previously investigated by Seöstrand by noticing that Gabor frames almost diagonalize the corresponding Weyl operators. This approach also allows to give new and more natural proofs of related results such as boundedness of operators or algebra and Wiener properties of the symbol class. Then, we discuss some recent…
The well know conjecture of {\it Coburn} [{\it L.A. Coburn, {On the Berezin-Toeplitz calculus}, Proc. Amer. Math. Soc. 129 (2001) 3331-3338.}] proved by {\it Lo} [{\it M-L. Lo, {The Bargmann Transform and Windowed Fourier Transform}, Integr. equ. oper. theory, 27 (2007), 397-412.}] and {\it Englis} [{\it M. Engli$\check{s}$, Toeplitz Operators and Localization Operators, Trans. Am. Math Society 361 (2009) 1039-1052.}] states that any {\it Gabor-Daubechies} operator with window $ψ$ and symbol ${\bf a}(x,ω)$ quantized on the phase space by a {\it Berezin-Toeplitz} operator with window $Ψ$ and sy…
We consider the Schrödinger equation \begin{equation*} i \displaystyle\frac{\partial u}{\partial t} +Hu=0,\quad H=a(x,D), \end{equation*} where the Hamiltonian $a(z)$, $z=(x,ξ)$, is assumed real-valued and smooth, with bounded derivatives $|\partial^αa(z)|\leq C_α$, for every $|α|\geq 2$, $z\in\mathbb{R}^{2d}$. For such equation results are known concerning well-posedness of the Cauchy problem for initial data in $L^2(\mathbb{R}^d)$ and local representation of the propagator $e^{it H}$ by means of Fourier integral operators. In the present paper we give a global expression for $e^{itH}$ in ter…
This paper introduces the concept of Bessel multipliers. These operators are defined by a fixed multiplication pattern, which is inserted between the Analysis and synthesis operators. The proposed concept unifies the approach used for Gabor multipliers for arbitrary analysis/synthesis systems, which form Bessel sequences, like wavelet or irregular Gabor frames. The basic properties of this class of operators are investigated. In particular the implications of summability properties of the symbol for the membership of the corresponding operators in certain operator classes are specified. As a s…
We obtain weighted $L^p$ inequalities for pseudo-differential operators with smooth symbols and their commutators by using a class of new weight functions which include Muckenhoupt weight functions. Our results improve essentially some well-known results.
We study decay and smoothness properties for eigenfunctions of compact localization operators. Operators with symbols a in the wide modulation space M^{p,\infty} (containing the Lebesgue space L^p), p<\infty, and windows \f_1,\f_2 in the Schwartz class are known to be compact. We show that their L^2-eigenfuctions with non-zero eigenvalues are indeed highly compressed onto a few Gabor atoms. Similarly, for symbols a in the weighted modulation spaces M^{\infty}_{v_s\otimes 1} (\rdd), s>0 (subspaces of M^{p,\infty}(\rdd), p>2d/s) the L^2-eigenfunctions of the localization operator are actually Sc…
Given a window $ϕ\in L^2(\mathbb R),$ and lattice parameters $α, β>0,$ we introduce a bimodal Wilson system $\mathcal{W}(ϕ, α, β)$ consisting of linear combinations of at most two elements from an associated Gabor $\mathcal{G}(ϕ, α, β)$. For a class of window functions $ϕ,$ we show that the Gabor system $\mathcal{G}(ϕ, α, β)$ is a tight frame of redundancy $β^{-1}$ if and only if the Wilson system $\mathcal{W}(ϕ, α, β)$ is Parseval system for $L^2(\mathbb R).$ Examples of smooth rapidly decaying generators $ϕ$ are constructed. In addition, when $3\leq β^{-1}\in \mathbb N$, we prove that it is …
We define new symbol classes for pseudodifferntial operators and investigate their pseudodifferential calculus. The symbol classes are parametrized by commutative convolution algebras. To every solid convolution algebra over a lattice we associate a symbol class. Then every operator with such a symbol is almost diagonal with respect to special wave packets (coherent states or Gabor frames), and the rate of almost diagonalization is described precisely by the underlying convolution algebra. Furthermore, the corresponding class of pseudodifferential operators is a Banach algebra of bounded opera…
We consider Fourier integral operators with symbols in modulation spaces and non-smooth phase functions whose second orders of derivatives belong to certain types of modulation space. We establish continuity and Schatten-von Neumann properties of such operators when acting on modulation spaces.
We consider Fourier integral operators with symbols in modulation spaces and non-smooth phase functions whose second orders of derivatives belong to certain types of modulation space. We establish continuity and Schatten-von Neumann properties of such operators when acting on modulation spaces.
We consider here pseudo-differential operators whose symbol $σ(x,ξ)$ is not infinitely smooth with respect to $x$. Decomposing such symbols into four -sometimes five- components and using tools of paradifferential calculus, we derive sharp estimates on the action of such pseudo-differential operators on Sobolev spaces and give explicit expressions for their operator norm in terms of the symbol $σ(x,ξ)$. We also study commutator estimates involving such operators, and generalize or improve the so-called Kato-Ponce and Calderon-Coifman-Meyer estimates in various ways.
Fourier integral operators with sufficiently smooth phase act on the time-frequency content of functions. However time-frequency analysis has only recently been used to analyze these operators. In this paper, we show that if a Fourier integral operator has a smooth phase function and its symbol is well-localized in time and frequency, then the operator is Schatten \( p \)-class for \( p \in [1,2] \), with inclusion of the symbol in mixed modulation spaces serving as the appropriate measure of time-frequency localization. Our main results are sharp in the sense that larger mixed modulation spac…
In this paper we prove sufficient conditions for the Fredholm property of a non-smooth pseudodifferential operator $P$ which symbol is in a Hölder space with respect to the spatial variable. As a main ingredient for the proof we use a suitable symbol-smoothing.
In this work, we develop $L^p$ boundedness theory for pseudodifferential operators with rough (not even continuous in general) symbols in the $x$ variable. Moreover, the $B(L^p)$ operator norms are estimated explicitly in terms of scale invariant quantities involving the symbols. All the estimates are shown to be sharp with respect to the required smoothness in the $ξ$ variable. As a corollary, we obtain $L^p$ bounds for (smoothed out versions of) the maximal directional Hilbert transform and the Carleson operator.
We study Mellin pseudodifferential operators (shortly, Mellin PDO's) with symbols in the algebra $\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$ of slowly oscillating functions of limited smoothness introduced in \cite{K09}. We show that if $\mathfrak{a}\in\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$ does not degenerate on the "boundary" of $\mathbb{R}_+\times\mathbb{R}$ in a certain sense, then the Mellin PDO ${\rm Op}(\mathfrak{a})$ is Fredholm on the space $L^p$ for $p\in(1,\infty)$ and each its regularizer is of the form ${\rm Op}(\mathfrak{b})+K$ where $K$ is a compact opera…
We consider special classes of linear bounded operators in Banach spaces and suggest certain operator variant of symbolic calculus. It permits to formulate an index theorem and to describe Fredholm properties of elliptic pseudo-differential operator on manifolds with non-smooth boundaries.
We investigate the sparsity of the Gabor-matrix representation of Fourier integral operators with a phase having quadratic growth. It is known that such an infinite matrix is sparse and well organized, being in fact concentrated along the graph of the corresponding canonical transformation. Here we show that, if the phase and symbol have a regularity of Gevrey type of order $s>1$ or analytic ($s=1$), the above decay is in fact sub-exponential or exponential, respectively. We also show by a counterexample that ultra-analytic regularity ($s<1$) does not give super-exponential decay. This is in s…
Smooth pseudodifferential operators on $\mathbb{R}^n$ can be characterized by their mapping properties between $L^p-$Sobolev spaces due to Beals and Ueberberg. In applications such a characterization would also be useful in the non-smooth case, for example to show the regularity of solutions of a partial differential equation. Therefore, we will show that every linear operator $P$, which satisfies some specific continuity assumptions, is a non-smooth pseudodifferential operator of the symbol-class $C^τ S^m_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$. The main new difficulties are the limited mapp…
Given a separable unital C*algebra $C$, let $E_n$ denote the Hilbert module equal to the completion of the Schwartz space of rapidly decreasing smooth functions from $R^n$ to $C$ equipped with the $C$-valued inner product given by integration. Let $B$ denote the space of all smooth functions with bounded derivatives from $R^n$ to $C$. For each $a$ in $B$, let $O(a)$ denote the pseudodifferential operator of symbol $a$. $O$ maps $B$ to $H$, the set of all adjointable operators on $E_n$ which have smooth orbit under the canonical action of the Heisenberg group. We construct a left inverse for $O…
We prove a higher order asymptotic formula for traces of finite block Toeplitz matrices with symbols belonging to Hölder-Zygmund spaces. The remainder in this formula goes to zero very rapidly for very smooth symbols. This formula refine previous asymptotic trace formulas by Szegö and Widom and complement higher order asymptotic formulas for determinants of finite block Toeplitz matrices due to Böttcher and Silbermann.
For a smooth manifold with boundary we construct a semigroupoid and a continuous field of C*-algebras which extend Connes' construction of the tangent groupoid. We show the asymptotic multiplicativity of \hbar-scaled truncated pseudodifferential operators with smoothing symbols and compute the K-theory of the associated symbol algebra.
We study the composition of time-ffrequency localization operators (wavepacket operators) and develop a symbolic calculus of such operators on modulation spaces. The use of time-frequency methods (phase space methods) allows the use of rough symbols of ultra-rapid growth in place of smooth symbols in the standard classes. As the main application it is shown that, in general, a localization operators possesses the Fredholm property, and thus its range is closed in the target space.
Suppose that $ϕ$ and $ψ$ are smooth complex-valued functions on the circle that are invertible, have winding number zero with respect to the origin, and have meromorphic extensions to an open neighborhood of the closed unit disk. Let $T_ϕ$ and $T_ψ$ denote the Toeplitz operators with symbols $ϕ$ and $ψ$ respectively. We give an explicit formula for the determinant of $T_ϕT_ψT_ϕ^{-1} T_ψ^{-1}$ in terms of the products of the tame symbols of $ϕ$ and $ψ$ on the open unit disk.
The characteristic functions of multivariate Feller processes with generator of affine type, and with smooth symbol functions have an explicit representation in terms of power series with rational number coefficients and with monmoms consisting of powers of the the symbol functions and formal derivatives of the symbol functions. The power series repesentations are convergent globally in time and on bounded domains of arbitrary size. Generalized symbol functions can be derived leading to power series expansions which are convergent on arbitrary domains in special cases. The rational number coef…
We investigate Fourier multipliers with smooth symbols defined over locally compact Hausdorff groups. Our main results in this paper establish new Hörmander-Mikhlin criteria for spectral and non-spectral multipliers. The key novelties which shape our approach are three. First, we control a broad class of Fourier multipliers by certain maximal operators in noncommutative $L_p$ spaces. This general principle ---exploited in Euclidean harmonic analysis during the last 40 years--- is of independent interest and might admit further applications. Second, we replace the formerly used cocycle dimensio…
We investigate truncated Toeplitz operators belonging to the Schatten ideals. We completely characterize such operators when they have an analytic symbol or belong to the ideal of Hilbert-Schmidt operators. We also study model spaces generated by Blaschke products associated with thin sequences, model spaces generated by certain types of singular inner functions, and operators associated with a class of very smooth symbols.
The main result in this paper is a one term Szego type asymptotic formula with a sharp remainder estimate for a class of integral operators of the pseudodifferential type with symbols which are allowed to be non-smooth or discontinuous in both position and momentum. The simplest example of such symbol is the product of the characteristic functions of two compact sets, one in real space and the other in momentum space. The results of this paper are used in a study of the violation of the area entropy law for free fermions in [18]. This work also provides evidence towards a conjecture due to Har…
In this paper, we investigate the mapping properties of pseudo-differential operators with operator-valued symbols. Thanks to the smooth atomic decomposition of the operator-valued Triebel-Lizorkin spaces $F_1^{α,c}(\mathbb{R}^d,\mathcal{M})$ obtained in our previous paper, we establish the $F_1^{α,c}$-regularity of regular symbols for every $α\in \mathbb{R}$, and the $F_1^{α,c}$-regularity of forbidden symbols for $α>0$. As applications, we obtain the same results on the usual and quantum tori.
In this paper we discuss some spectral invariance results for non-smooth pseudodifferential operators with coefficients in Hölder spaces. In analogy to the proof in the smooth case of Beals and Ueberberg, we use the characterization of non-smooth pseudodifferential operators to get such a result. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols and the fact, that in general the composition of two non-smooth pseudodifferential operators is not a pseudodifferential operator. In order to improve these spectral invariance results …
