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1.
Let L   be a non-negative self-adjoint operator acting on L2(X)L2(X) where X is a space of homogeneous type. Assume that L   generates a holomorphic semigroup e−tLetL 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.  相似文献   

2.
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 Mω,1, then the matrix has polynomial or exponential off-diagonal decay, depending on the weight ω. Moreover, if its operator is invertible on L2, 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.  相似文献   

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Let Φ and Ψ be frames for ? and let Mm,Φ,Ψ be a frame multiplier with the symbol m. In this paper, we restrict our investigation to show that the operator properties of Mm,Φ,Ψ are stable under the perturbations of Φ, Ψ, and m. Also, special attention is devoted to the study of invertible frame multipliers.  相似文献   

5.
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.  相似文献   

6.
We consider a class of linear Schrödinger equations in RdRd, 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.  相似文献   

7.
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.  相似文献   

8.
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 L2(Rd). We obtain an MOEP for (non)homogeneous affine dual frames and (non)homogeneous affine Parseval frames.  相似文献   

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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 L2(R) to the vector-valued Hilbert space L2(R,CL), 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 L2(R,CL). Furthermore, we show the existence of painless vector-valued NSG frames, and of vector-valued NSG frames with fast decaying window functions.  相似文献   

11.
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 R d ? { 0 } . We construct simple adapted tight frames for L 2 ( R d ) 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.  相似文献   

12.
Inspired by results of Kim and Ron, given a Gabor frame in L2(R), we determine a non-countable generalized frame for the non-separable space AP2(R) of the Besicovic almost periodic functions. Gabor type frames for suitable separable subspaces of AP2(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.  相似文献   

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We show that (g2,a,b) is a Gabor frame when a>0, b>0, ab<1, and g2(t)=(12πγ)1/2(coshπγt)−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.  相似文献   

16.
In this paper we obtain conditions under which the operator equations of the types AX = C and AXA1 = 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.  相似文献   

17.
A finitely generated shift invariant space V   is a closed subspace of L2(Rd)L2(Rd) 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.  相似文献   

18.
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.  相似文献   

19.
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.  相似文献   

20.
在框架理论研究中,哪类可逆算子能使得某些框架性质保持不变这个问题是基本和重要的,本文在无穷维Hilbert空间上对下述两个问题进行研究.问题1:哪类可逆算子能使得框架算子保持不变;问题2:哪类可逆算子能使得框架范数只相差一列常数.本文从抽象的算子理论和具体的构造方法两方面对问题1给出解答.利用框架的相容算子的概念,当把问题2中的可逆算子集换成一类较小的算子集时,得到了问题2的回答.  相似文献   

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