Multi-window Gabor frames and oblique Gabor duals on discrete periodic sets

Given L, N, M ∈ N and an NZ-periodic set S in Z, let l2(S) be the closed subspace of l2(Z) consisting of sequences vanishing outside S. For f = { fl : 0≤l≤L-1 }l2(Z), we denote by G(f, N, M) the Gabor system generated by f, and by L(f, N, M) the closed linear subspace generated by G(f, N, M). This paper addresses density results, frame conditions for a Gabor system G(g, N, M) in l2(S), and Gabor duals of the form G(a, N, M) in some L(h, N, M) for a frame G(g, N, M) in l2(S) (so-called oblique duals). We ob...     

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).     

Estimating reachable sets for two-dimensional linear discrete systems

We are concerned with the problem of estimating the reachable set for a two-dimensional linear discrete-time system with bounded controls. Different approaches are adopted depending on whether the system matrix has real or complex eigenvalues. For the complex eigenvalue case, the quasiperiodic nature of minimum time trajectories is exploited in developing a simple, but often accurate, procedure. For the real eigenvalue case, over estimates of reachable sets can be trivially obtained using a decomposition method.The second author was supported by funds supplied by the John M. Bennett Faculty Fellowship, Trinity University, San Antonio, Texas.     

Density results for subspace multiwindow Gabor systems in the rational case

Let S be a periodic set in R and L2(S) be a subspace of L2 (R). This paper investigates the density problem for multiwindow Gabor systems in L2(S) for the case that the product of time-frequency shift parameters is a rational number. We derive the density conditions for a multiwindow Gabor system to be complete (a frame) in L2(S). Under such conditions, we construct a multiwindow tight Gabor frame for L2(S) with window functions being characteristic functions. We also provide a characterization of a multiwindow Gabor frame to be a Riesz basis for L2(S), and obtain the density condition for a multiwindow Gabor Riesz basis for L2(S).     

Time-domain characterization of multiwindow Gabor systems on discrete periodic sets

This paper addresses multiwindow Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. We give some necessary and/or sufficient conditions for multiwindow Gabor systems to foe frames on discrete periodic sets, and characterize two multiwindow Gabor Bessel sequences to foe dual frames on discrete periodic sets. For a given multiwindow Gabor frame, we derive all its Gabor duals, among which we obtain an explicit expression of the canonical Gabor dual. In addition, we generalize multiwindow Gabor systems to the case of a different sampling rate for each window, and investigate multiwindow Gabor frames and dual frames in this case. We also show the properties of the multiwindow Gabor systems are essentially not changed by replacing the exponential kernel with other kernels.     

Super Gabor frames on discrete periodic sets

Due to its potential applications in multiplexing techniques such as time division multiple access and frequency division multiple access, superframe has interested some mathematicians and engineering specialists. In this paper, we investigate super Gabor systems on discrete periodic sets in terms of a suitable Zak transform matrix, which can model signals to appear periodically but intermittently. Complete super Gabor systems, super Gabor frames and Gabor duals for super Gabor frames on discrete periodic sets are characterized; An explicit expression of Gabor duals is established, and the uniqueness of Gabor duals is characterized. On the other hand, discrete periodic sets admitting complete super Gabor systems, super Gabor frames, super Gabor Riesz bases are also characterized. Some examples are also provided to illustrate the general theory.     

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.     

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.     

Optimal steady-state versus periodic operation in discrete systems

The problem of improving the optimal steady state of a discrete system by means of a suitable cyclic operation is discussed. Two sufficient conditions are found by negating a necessary condition and by following a frequency-domain approach, respectively. The connection between the two conditions is also shown.This work has been supported by the Consiglio Nazionale delle Ricerche, Rome, Italy.     

Existence of periodic solutions for a class of second-order discrete Hamiltonian systems

By using the variant fountain theorem, we study the existence of periodic solutions for a class of superquadratic non-autonomous second-order discrete Hamiltonian systems.     

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.     

Front propagation for discrete periodic monostable equations

This paper deals with front propagation for discrete periodic monostable equations. We show that there is a minimal wave speed such that a pulsating traveling front solution exists if and only if the wave speed is above this minimal speed. Moreover, in comparing with the continuous case, we prove the convergence of discretized minimal wave speeds to the continuous minimal wave speed.     

Wavelet decomposition of the space of discrete periodic splines

An orthogonal basis for the spaceS _r ^m of discrete periodic splines is constructed. The wavelet decomposition of the spaceS _r ^m form=2 ^t is obtained using this basis. We derive recurrence formulas for the transformation from the decomposition with respect to the orthogonal basis to the wavelet decomposition, as well as recurrence formulas for the inverse transformation. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 712–720, May, 2000.     

Applications of Clark duality to periodic solutions with minimal period for discrete Hamiltonian systems

By making use of Clark duality, perturbation technique and dual least action principle, some results on the existence of subharmonic solutions with minimal period to second-order subquadratic discrete Hamiltonian systems are obtained.     

Nonstationary Gabor frames for

Recently, continuous‐time nonstationary Gabor (NSG) frames were introduced in adaptive signal analysis. They allow for efficient reconstruction with flexible sampling and varying window functions. In this paper, we focus on the existence and construction of NSG frames in the discrete‐time setting. We provide existence results for painless NSG frames and for NSG frames with fast decaying window functions. We also construct NSG frames with noncompactly supported window functions from a known painless NSG frame. Some examples are provided to illustrate the general theory.     

Almost periodic transformable difference systems

We analyse almost periodic homogeneous linear difference systems whose coefficient matrices belong to the so-called transformable groups. We introduce the notion of weakly transformable groups and we use this notion to obtain generalizations of known results about non-almost periodic solutions of considered systems.     

Global periodicity and complete integrability of discrete dynamical systems

Consider the discrete dynamical system generated by a map F. It is said that it is globally periodic if there exists a natural number p such that F ^p (x)=x for all x in the phase space. On the other hand, it is called completely integrable if it has as many functionally independent first integrals as the dimension of the phase space. In this paper, we relate both concepts. We also give a large list of globally periodic dynamical systems together with a complete set of their first integrals, emphasizing the ones coming from difference equations.     

Almost periodic homogeneous linear difference systems without almost periodic solutions

Almost periodic homogeneous linear difference systems are considered. It is supposed that the coefficient matrices belong to a group. The aim was to find such groups that the systems having no non-trivial almost periodic solution form a dense subset of the set of all considered systems. A closer examination of the used methods reveals that the problem can be treated in such a generality that the entries of coefficient matrices are allowed to belong to any complete metric field. The concepts of transformable and strongly transformable groups of matrices are introduced, and these concepts enable us to derive efficient conditions for determining what matrix groups have the required property.