Sharp spectral multipliers for operators satisfying generalized Gaussian estimates

Let L   be a non-negative self-adjoint operator acting on L²(X)$L^2(X)$ where X is a space of homogeneous type. Assume that L   generates a holomorphic semigroup e^−tL$e^{-tL}$ which satisfies generalized m-th order Gaussian estimates. In this article, we study singular and dyadically supported spectral multipliers for abstract self-adjoint operators. We show that in this setting sharp spectral multiplier results follow from Plancherel or Stein–Tomas type estimates. These results are applicable to spectral multipliers for a large class of operators including m-th order elliptic differential operators with constant coefficients, biharmonic operators with rough potentials and Laplace type operators acting on fractals.     

Gabor discretization of the Weyl product for modulation spaces and filtering of nonstationary stochastic processes

We discretize the Weyl product acting on symbols of modulation spaces, using a Gabor frame defined by a Gaussian function. With one factor fixed, the Weyl product is equivalent to a matrix multiplication on the Gabor coefficient level. If the fixed factor belongs to the weighted Sjöstrand space $M_{\omega }^{\infty ,1}$, then the matrix has polynomial or exponential off-diagonal decay, depending on the weight ω. Moreover, if its operator is invertible on $L^2$, the inverse matrix has similar decay properties. The results are applied to the equation for the linear minimum mean square error filter for estimation of a nonstationary second-order stochastic process from a noisy observation. The resulting formula for the Gabor coefficients of the Weyl symbol for the optimal filter may be interpreted as a time–frequency version of the filter for wide-sense stationary processes, known as the noncausal Wiener filter.     

Invariances of the Operator Properties of Frame Multipliers Under Perturbations of Frames and Symbol

Let Φ and Ψ be frames for ? 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.     

Representation of Operators in the Time-Frequency Domain and Generalized Gabor Multipliers

Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A characterization of operators that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator’s best approximation by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error estimates for these approximations. Generalizations (multiple Gabor multipliers) are introduced for better approximation of overspread operators. The Riesz property of the projection operators involved in generalized Gabor multipliers is characterized, and a method for obtaining an operator’s best approximation by a multiple Gabor multiplier is suggested. Finally, it is shown that in certain situations, generalized Gabor multipliers reduce to a finite sum of regular Gabor multipliers with adapted windows.     

Wave packet analysis of Schrödinger equations in analytic function spaces

We consider a class of linear Schrödinger equations in R^d${\mathbb{R}}^d$, 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 singularities.     

Basic definition and properties of Bessel multipliers

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 special case the multipliers for Riesz bases are examined and it is shown that multipliers in this case can be easily composed and inverted. Finally the continuous dependence of a Bessel multiplier on the parameters (i.e., the involved sequences and the symbol in use) is verified, using a special measure of similarity of sequences.     

Extension principles for affine dual frames in reducing subspaces

Mixed Oblique Extension Principles (MOEP) provide an important method to construct affine dual frames from refinable functions. This paper addresses MOEP under the setting of reducing subspaces of $L^2({\mathbb{R}}^d)$. We obtain an MOEP for (non)homogeneous affine dual frames and (non)homogeneous affine Parseval frames.     

Vector-valued nonstationary Gabor frames

As an extension of Gabor frames, nonstationary Gabor (NSG) frames were recently introduced in adaptive signal analysis. They allow for efficient reconstruction with flexible sampling and varying window functions. In this paper we generalize the notion of NSG frames from $L^2(\mathbb{R})$ to the vector-valued Hilbert space $L^2(\mathbb{R},{\mathbb{C}}^L)$, and investigate the resulting vector-valued NSG frames. We derive a Walnut's representation of the mixed frame operator, and provide some necessary/sufficient conditions for a vector-valued NSG system to be a frame for $L^2(\mathbb{R},{\mathbb{C}}^L)$. Furthermore, we show the existence of painless vector-valued NSG frames, and of vector-valued NSG frames with fast decaying window functions.     

On homogeneous decomposition spaces and associated decompositions of distribution spaces

A new construction of decomposition smoothness spaces of homogeneous type is considered. The smoothness spaces are based on structured and flexible decompositions of the frequency space ${\mathbb{R}}^d?\{0\}$. We construct simple adapted tight frames for $L_2\left({\mathbb{R}}^d\right)$ that can be used to fully characterise the smoothness norm in terms of a sparseness condition imposed on the frame coefficients. Moreover, it is proved that the frames provide a universal decomposition of tempered distributions with convergence in the tempered distributions modulo polynomials. As an application of the general theory, the notion of homogeneous α‐modulation spaces is introduced.     

Gabor systems and almost periodic functions

Inspired by results of Kim and Ron, given a Gabor frame in $L^2(\mathbb{R})$, we determine a non-countable generalized frame for the non-separable space ${\mathit{AP}}_2(\mathbb{R})$ of the Besicovic almost periodic functions. Gabor type frames for suitable separable subspaces of ${\mathit{AP}}_2(\mathbb{R})$ are constructed. We show furthermore that Bessel-type estimates hold for the AP norm with respect to a countable Gabor system using suitable almost periodic norms of sequences.     

Hyperbolic Secants Yield Gabor Frames

We show that (g₂,a,b) is a Gabor frame when a>0, b>0, ab<1, and $\text{g}{}_2\text{(t)=(}\text{1}\text{2}\text{πγ)}{}^{\mathrm{1/2}}\text{(cosh}\text{πγt)}{}^{\mathrm{-1}}$ is a hyperbolic secant with scaling parameter γ>0. This is accomplished by expressing the Zak transform of g₂ in terms of the Zak transform of the Gaussian g₁(t)=(2γ)^1/4 exp (−πγt²), together with an appropriate use of the Ron–Shen criterion for being a Gabor frame. As a side result it follows that the windows, generating tight Gabor frames, that are canonically associated to g₂ and g₁ are the same at critical density a=b=1. Also, we display the “singular” dual function corresponding to the hyperbolic secant at critical density.     

Generalized inverses and operator equations

In this paper we obtain conditions under which the operator equations of the types AX = C and $\text{AXA}{}^1\text{= C}$ have hermitian and nonnegative definite solutions; here A is assumed to be relatively invertible. In addition we obtain some properties of generalized inverses of operators. Lastly we pose some conjectures; one of them is that the set of all nonzero relatively invertible operators is not connected.     

Linear combinations of frame generators in systems of translates

A finitely generated shift invariant space V   is a closed subspace of L²(R^d)$L^2({\mathbb{R}}^d)$ that can be generated by the integer translates of a finite number of functions. A set of frame generators for V is a set of functions whose integer translates form a frame for V. In this note we give necessary and sufficient conditions in order that a minimal set of frame generators can be obtained by taking linear combinations of the given frame generators. Surprisingly the results are very different from the recently studied case when the property to be a frame is not required.     

Pair frames in Hilbert $$\varvec{C^*}$$-modules

In this paper, we introduce pair frames in Hilbert \(C^*\)-modules and show that they share many useful properties with their corresponding notions in Hilbert spaces. We also obtain the necessary and sufficient conditions for a standard Bessel sequence to construct a pair frame and get the necessary and sufficient conditions for a Hilbert \(C^*\)-module to admit a pair frame with a symbol and two standard Bessel sequences. Moreover by generalizing some of the results obtained for Bessel multipliers in Hilbert \(C^*\)-modules to pair frames and considering the stability of pair frames under invertible operators, we construct new pair frames and show that pair frames are stable under small perturbations.     

Generalized super Gabor duals with bounded invertible operators

In this paper, we introduce generalized super Gabor duals with bounded invertible operators by combining ideas concerning super Gabor frames with the idea of g-duals as proposed by Dehgham and Fard in 2013. Given a super Gabor frame and a bounded invertible operator A, we characterize its generalized super Gabor duals with A, and derive a parametric expression of all its generalized super Gabor duals with A. The perturbation of generalized super Gabor duals is considered as well.