Density results for Gabor systems associated with periodic subsets of the real line

The well-known density theorem for one-dimensional Gabor systems of the form , where , states that a necessary and sufficient condition for the existence of such a system whose linear span is dense in , or which forms a frame for , is that the density condition is satisfied. The main goal of this paper is to study the analogous problem for Gabor systems for which the window function g vanishes outside a periodic set which is -shift invariant. We obtain measure-theoretic conditions that are necessary and sufficient for the existence of a window g such that the linear span of the corresponding Gabor system is dense in L²(S). Moreover, we show that if this density condition holds, there exists, in fact, a measurable set with the property that the Gabor system associated with the same parameters a,b and the window g=χ_E, forms a tight frame for L²(S).     

Gabor systems on discrete periodic sets

Due to its good potential for digital signal processing, discrete Gabor analysis has interested some mathematicians. This paper addresses Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. Complete Gabor systems and Gabor frames on discrete periodic sets are characterized; a sufficient and necessary condition on what periodic sets admit complete Gabor systems is obtained; this condition is also proved to be sufficient and necessary for the existence of sets E such that the Gabor systems generated by χ _E are tight frames on these periodic sets; our proof is constructive, and all tight frames of the above form with a special frame bound can be obtained by our method; periodic sets admitting Gabor Riesz bases are characterized; some examples are also provided to illustrate the general theory. This work was supported by National Natural Science Foundation of China (Grant No. 10671008), Beijing Natural Science Foundation (Grant No. 1092001), PHR (IHLB) and the project sponsored by SRF for ROCS, SEM of China     

Density of Gabor Frames

A Gabor system is a set of time-frequency shifts S(g, Λ) ={e^{2 π ibx}g(x − a)}_{(a, b) Λ} of a function g L²(R^d). We prove that if a finite union of Gabor systems _{k = 1}^rS(g_k, Λ_k) forms a frame for L²(R^d) then the lower and upper Beurling densities of Λ = _{k = 1}^r Λ_k satisfy D⁻(Λ) ≥ 1 and D ⁺ (Λ) < ∞. This extends recent work of Ramanathan and Steger. Additionally, we prove the conjecture that no collection _{k = 1}^r{g_k(x − a)}_{a Γk} of pure translates can form a frame for L²(R^d).     

The existence of tight Gabor duals for Gabor frames and subspace Gabor frames

Let K and L be two full-rank lattices in Rd. We give a complete characterization for all the Gabor frames that admit tight dual of the same type. The characterization is given in terms of the center-valued trace of the von Neumann algebra generated by the left regular projective unitary representations associated with the time-frequency lattice K×L. Two applications of this characterization were obtained: (i) We are able to prove that every Gabor frame has a tight dual if and only if the volume of K×L is less than or equal to . (ii) We are able to obtain sufficient or necessary conditions for the existence of tight Gabor pseudo-duals for subspace Gabor frames in various cases. In particular, we prove that every subspace Gabor frame has a tight Gabor pseudo-dual if either the volume or v(K×L)?2. Moreover, if K=αZd, L=βZd with αβ=1, then a subspace Gabor frame G(g,L,K) has a tight Gabor pseudo-dual only when G(g,L,K) itself is already tight.     

Sufficient conditions for irregular Gabor frames

Finding general and verifiable conditions which imply that Gabor systems are (resp. cannot be) Gabor frames is among the core problems in Gabor analysis. In their paper on atomic decompositions for coorbit spaces [H.G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations, and their atomic decomposition, I, J. Funct. Anal. 86 (1989), 307–340], the authors proved that every Gabor system generated with a relatively uniformly discrete and sufficiently dense time-frequency sequence will allow series expansions for a large class of Banach spaces if the window function is nice enough. In particular, such a Gabor system is a frame for the Hilbert space of square integrable functions. However, their proof is based on abstract analysis and does not give direct information on how to determine the density in the sense of directly applicable estimates. It is the goal of this paper to present a constructive version of the proof and to provide quantitative results. Specifically, we give a criterion for the general case and explicit density for some cases. We also study the existence of Gabor frames and show that there is some smooth window function such that the corresponding Gabor system is incomplete for arbitrary time-frequency lattices.     

Relaxed quaternionic Gabor expansions at critical density

Shifted and modulated Gaussian functions play a vital role in the representation of signals. We extend the theory into a quaternionic setting, using two exponential kernels with two complex numbers. As a final result, we show that every continuous and quaternion‐valued signal f in the Wiener space can be expanded into a unique ? ² series on a lattice at critical density 1, provided one more point is added in the middle of a cell. We call that a relaxed Gabor expansion . Copyright © 2016 John Wiley & Sons, Ltd.     

Weighted irregular Gabor tight frames and dual systems using windows in the Schwartz class

We give a characterization for the weighted irregular Gabor tight frames or dual systems in L²(Rn) in terms of the distributional symplectic Fourier transform of a positive Borel measure on R2n naturally associated with the system and the short-time Fourier transform of the windows in the case where the window (or at least one of the windows in the case of dual systems) belongs to S(Rn). This result implies that, for certain classes of windows such as generalized Gaussians or “extreme-value” windows, the only weighted irregular Gabor tight frames (or even dual systems with both windows in the same class) that can be constructed with these windows are the trivial ones, corresponding to the measure μ=1 on R2n. Furthermore, we show that, if a such Gabor system admits a dual which is of Gabor type, then the Beurling density of the associated measure exists and is equal to one.     

Recent computational developments in Krylov subspace methods for linear systems

Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters. Copyright © 2006 John Wiley & Sons, Ltd.     

Beurling Dimension of Gabor Pseudoframes for Affine Subspaces

Pseudoframes for subspaces have been recently introduced by Li and Ogawa as a tool to analyze lower dimensional data with arbitrary flexibility of both the analyzing and the dual sequence. In this paper we study Gabor pseudoframes for affine subspaces by focusing on geometrical properties of their associated sets of parameters. We first introduce a new notion of Beurling dimension for discrete subsets of ℝ ^d by employing a certain generalized Beurling density. We present several properties of Beurling dimension including a comparison with other notions of dimension showing, for instance, that our notion includes the mass dimension as a special case. Then we prove that Gabor pseudoframes for affine subspaces satisfy a certain Homogeneous Approximation Property, which implies invariance under time–frequency shifts of an approximation by elements from the pseudoframe. The main result of this paper is a classification of Gabor pseudoframes for affine subspaces by means of the Beurling dimension of their sets of parameters. This provides us, in particular, with a Nyquist dimension which separates sets of parameters of pseudoframes from those of non-pseudoframes and which links a fixed value to sets of parameters of pseudo-Riesz sequences. These results are even new for the special case of Gabor frames for an affine subspace.      

On the maximal dimension of a completely entangled subspace for finite level quantum systems

LetH _ibe a finite dimensional complex Hilbert space of dimensiond _i associated with a finite level quantum system A_i for i = 1, 2, ...,k. A subspaceS ⊂ is said to becompletely entangled if it has no non-zero product vector of the formu ₁⊗u ₂ ⊗ ... ⊗u _k with u_i inH _i for each i. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic algebraic geometry we prove that

Global results on rational systems in the plane,part 1

Edited by Gerry Ladas

In this section we present some open problems and conjectures about some interesting types of difference equations. Please submit your problems and conjectures with all relevant information to G. Ladas: Email: gladas@math.uri.edu     

Calderón-Zygmund operators on amalgam spaces and in the discrete case

We prove the boundedness of Calderón-Zygmund operators on weighted amalgam spaces for 1<p,q<∞ with Muckenhoupt weights. To do this, we show the boundedness in the discrete case, i.e. the boundedness on . We also investigate on . As an application we consider an operator related to the Navier-Stokes equation.     

Optimal shape design for systems governed by variational inequalities,part 3: Necessary conditions in the elliptic case

General optimality conditions are obtained for optimal shape design for systems governed by a class of elliptic variational inequalities. The conditions are established by making use of the necessary conditions for optimal control of systems governed by strongly monotone variational inequalities. These conditions are then applied to an electrochemical machining problem.     

Controllability results for the two-dimensional heat equation with mixed boundary conditions using Carleman inequalities: a linear and a semilinear case

In this paper, we prove controllability results for a two-dimensional semilinear heat equation with mixed boundary conditions. It is well-known that mixed boundary conditions can present a singular behaviour of the solution. First, we will prove global Carleman estimates then we will use these inequalities to obtain controllability results.     

Partial regularity results for subelliptic systems in the Heisenberg group

We consider subelliptic systems in the Heisenberg group. We give a new proof for the smoothness of solutions of inhomogeneous systems with constant coefficients. With this result, we prove partial Hölder continuity of the horizontal gradient for non-linear systems with p-growth for p≥2 via the $\mathcal {A}We consider subelliptic systems in the Heisenberg group. We give a new proof for the smoothness of solutions of inhomogeneous systems with constant coefficients. With this result, we prove partial H?lder continuity of the horizontal gradient for non-linear systems with p-growth for p≥2 via the -harmonic approximation technique.     

Symmetry results for decay solutions of elliptic systems in the whole space

In this paper, by using the Alexandrov-Serrin method of moving planes combined with maximum principles, we prove that the decaying positive solutions of a semi-linear elliptic system in the whole space are radially symmetric about some point. The system under our consideration includes the important physical interesting case, the stationary Schrödinger system for Bose-Einstein condensate.     

Asymptotic Expansion of the Distribution Density Function for the Sum of Random Variables in the Series Scheme in Large Deviation Zones

The work is designated for obtaining asymptotic expansions and determination of structures of the remainder terms that take into consideration large deviations both in the Cramer zone and Linnik power zones for the distribution density function of sums of independent random variables in a triangular array scheme. The result was obtained using general Lemma 6.1 of Saulis and Statuleviius in Limit Theorems for Large Deviations (Kluwer, 1991) and joining the methods of characteristic functions and cumulants. The work extends the theory of sums of random variables and in a special case, improves S. A.Book's results on sums of random variables with weights.     

On a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case II

The main theme of this paper is the discussion of a family of extremal solutions of a finite moment problem for rational matrix functions in the nondegenerate case. We will point out that each member of this family is extremal in several directions. Thereby, the investigations below continue the studies in Fritzsche et al. (in press) [1]. In doing so, an application of the theory of orthogonal rational matrix functions with respect to a nonnegative Hermitian matrix Borel measure on the unit circle is used to get some insights into the structure of the extremal solutions in question. In particular, we explain characterizations of these solutions in the whole solution set in terms of orthogonal rational matrix functions. We will also show that the associated Riesz-Herglotz transform of such a particular solution admits specific representations, where orthogonal rational matrix functions are involved.     

The generalized Cauchy problem for singularly perturbed impulsive systems in the critical case

We construct an asymptotic expansion for solutions to nonlinear singularly perturbed systems of impulsive differential equations. We successively determine all terms of the asymptotic expansion by means of pseudoinverse matrices and orthoprojections.     

Global existence of classical solutions to the Cauchy problem for a kind of quasilinear hyperbolic systems in divergence form

This paper deals with the global existence of classical solutions to a kind of second order quasilinear hyperbolic systems subject to a null condition, with the linear elastodynamic system as its principal part and the nonlinear terms depending on the product of u² and the derivatives of u.