Sampling set conditions for shift-invariant signal spaces

We study nonuniform sampling in shift-invariant spaces and the construction of Gabor frames with respect to the class of totally positive functions whose Fourier transform factors as hat{g}(xi )= prod _{j=1}^n (1+2pi idelta _jxi )^{-1} , e^{-c xi ^2} for delta _1,ldots ,delta _nin mathbb {R}, c >0 (in which case g is called totally positive of Gaussian type). In analogy to Beurling’s sampling theorem for the Paley–Wiener space of entire functions, we prove that every separated set with lower Beurling density >1 is a sampling set for the shift-invariant space generated by such a g. In view of the known necessary density conditions, this result is optimal and validates the heuristic reasonings in the engineering literature. Using a subtle connection between sampling in shift-invariant spaces and the theory of Gabor frames, we show that the set of phase-space shifts of g with respect to a rectangular lattice alpha mathbb {Z}times beta mathbb {Z} forms a frame, if and only if alpha beta <1. This solves an open problem going back to Daubechies in 1990 for the class of totally positive functions of Gaussian type. The proof strategy involves the connection between sampling in shift-invariant spaces and Gabor frames, a new characterization of sampling sets “without inequalities” in the style of Beurling, new properties of totally positive functions, and the interplay between zero sets of functions in a shift-invariant space and functions in the Bargmann–Fock space.

Recovery of compressive samples from a randomly modulated converter

The aim of this paper is to study sampling and reconstruction in a shift invariant space with multiple generators. On contrary to the classical case of a shift invariant space with a single generator, it is shown that cannot be a stable set of sampling for where when Further the problems of perturbation of a stable set of sampling and local reconstruction method are discussed along with an illustration and implementation.

Extra invariance of shift-invariant spaces on LCA groups

In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

In many applications only the low frequency components of a signal can be measured due to the lowpass behavior of many physical systems. Nevertheless, if additional information on the structure of the signal is known, it might still be possible to reconstruct the signal from its low-frequency content. This paper studies signals in shift-invariant spaces with multiple generators and derives necessary conditions on the bandwith of the lowpass filter as well as sufficient conditions on the generators such that signal recovery is possible. If the signal can not be recovered from its low frequency components, an appropriate pre-processing of the signal is proposed which improves the reconstruction ability. In particular, it is shown that modulating the signal with one or more mixing functions prior to lowpass filtering can ensure the recovery of the signal in many cases.

Let ? be a function in the Wiener amalgam space $\emph{W}_{\infty}(L_1)$ with a non-vanishing property in a neighborhood of the origin for its Fourier transform $\widehat{\phi}$ , ${\bf \tau}=\{\tau_n\}_{n\in {{\mathbb Z}}}$ be a sampling set on ? and $V_\phi^{\bf \tau}$ be a closed subspace of $L_2(\hbox{\ensuremath{\mathbb{R}}})$ containing all linear combinations of ?-translates of ?. In this paper we prove that every function $f\in V_\phi^{\bf \tau}$ is uniquely determined by and stably reconstructed from the sample set $L_\phi^{\bf \tau}(f)=\Big\{\int_{\hbox{\ensuremath{\mathbb{R}}}} f(t) \overline{\phi(t-\tau_n)} dt\Big\}_{n\in {{\mathbb Z}}}$ . As our reconstruction formula involves evaluating the inverse of an infinite matrix we consider a partial reconstruction formula suitable for numerical implementation. Under an additional assumption on the decay rate of ? we provide an estimate to the corresponding error.

EXPOSE-R flew as the second of the European Space Agency (ESA) EXPOSE multi-user facilities on the International Space Station. During the mission on the external URM-D platform of the Zvezda service module, samples of eight international astrobiology experiments selected by ESA and one Russian guest experiment were exposed to low Earth orbit space parameters from March 10th, 2009 to January 21st, 2011. EXPOSE-R accommodated a total of 1220 samples for exposure to selected space conditions and combinations, including space vacuum, temperature cycles through 273 K, cosmic radiation, solar electromagnetic radiation at &gt;110, &gt;170 or &gt;200 nm at various fluences up to GJ m−2. Samples ranged from chemical compounds via unicellular organisms and multicellular mosquito larvae and seeds to passive radiation dosimeters. Additionally, one active radiation measurement instrument was accommodated on EXPOSE-R and commanded from ground in accordance with the facility itself. Data on ultraviolet radiation, cosmic radiation and temperature were measured every 10 s and downlinked by telemetry and data carrier every few months. The EXPOSE-R trays and samples returned to Earth on March 9th, 2011 with Shuttle flight, Space Transportation System (STS)-133/ULF 5, Discovery, after successful total mission duration of 27 months in space. The samples were analysed in the individual investigators laboratories. A parallel Mission Ground Reference experiment was performed on ground with a parallel set of hardware and samples under simulated space conditions following to the data transmitted from the flight mission.

This paper systematically studies the existence of maximal elements for unordered binary relations on compact or noncompact sets in a general topological space. This is done by developing a method, called transfer method, to derive various necessary and sufficient conditions that characterize the existence of maximal elements for a binary relation in terms of: (1) (generalized) transitivity conditions under certain topological assumptions; (2) topological conditions under certain (generalized) transitivity assumptions; and (3) (generalized) convexity conditions under certain topological assumptions. There are two basic approaches in the literature to prove the existence by providing sufficient conditions. One assumes certain convexity and continuity conditions for a topological vector space and the other assumes certain weakened transitivity and continuity conditions for a general topological space. The results unify those two aproaches and generalize almost all of the existing results in the literature.

We propose necessary and sufficient conditions for the local Morrey space to have one of the following properties: the absolutely continuous norm, the Fatou property, the reflexivity. We propose conditions for the global Morrey space \(GM_ {l, X}^\tau \) to have the Fatou property. We give an example of global Morrey spaces \( GM_{l,X}^ \tau \), which do have not an absolutely continuous norm, and spaces l and X have an absolutely continuous norm.

Methods for the calculation of the so-called dual Gabor window for general sampling sets are developed. First, we consider the standard situation where the time-frequency plane (TF-plane) is (over)sampled on a grid, i.e., a rectangular lattice is considered with more points than the dimension of the signal space. Second, we show that the proposed method can be easily extended to general subgroups of the TF-plane. One typical example of such a sampling set is the so called quincunx-lattice.

Zero-reflectivity surfaces on lossy materials are useful for eliminating unwanted reflections. Surface-relief gratings on metal surfaces can produce unity transmissivity and thus total absorption. A rigorous impedance matching approach is used. The zero reflectivity condition and the equivalence of the homogeneous layer and the grating in the long-wavelength limit produce a system of nonlinear transcendental equations. First, the required thickness and complex refractive index for a single homogeneous lossy layer antireflection coating are calculated. Then the filling factor and groove depth of the equivalent high-spatial-frequency rectangular-groove surface-relief grating are determined. The analysis includes multiple dielectric overlayers for grating protection and for broad banding the antireflection characteristics. It is shown that multiple zero reflectivity solutions exist for both TE and TM polarizations for arbitrary angles of incidence and for any complex refractive-index substrate. Necessary and sufficient conditions in the parametric space are also computed. The inclusion of one or more dielectric overlayers gives additional design flexibility. The sensitivity of the antireflection design as a function of the angle of incidence and wavelength is presented.

Let N be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra \(\mathfrak{n}\) having rational structure constants. We assume that \(N = P \rtimes M,\) M is commutative, and for all \(\lambda \in \mathfrak{n}^{{\ast}}\) in general position the subalgebra \(\mathfrak{p} =\log (P)\) is a polarization ideal subordinated to λ (\(\mathfrak{p}\) is a maximal ideal satisfying \([\mathfrak{p},\mathfrak{p}] \subseteq \ker \lambda\) for all λ in general position and \(\mathfrak{p}\) is necessarily commutative.) Under these assumptions, we prove that there exists a discrete uniform subgroup Γ ⊂ N such that L2(N) admits band-limited spaces with respect to the group Fourier/Plancherel transform which are sampling spaces with respect to Γ. We also provide explicit sufficient conditions which are easily checked for the existence of sampling spaces. Sufficient conditions for sampling spaces which enjoy the interpolation property are also given. Our result bears a striking resemblance with the well-known Whittaker-Kotel’nikov-Shannon sampling theorem.

Fractional integro-differential equations arise in the mathematical modeling of various physical phenomena like heat conduction in materials with memory, diffusion processes etc. In this article, sufficient conditions are derived for approximate controllability of impulsive fractional integro-differential systems with nonlocal conditions in Hilbert space. The results are obtained by using fractional calculus, semigroup theory and the Darbo-Sadovskii's fixed point theorem. For an application, a specific type of ultraslow diffusion type of porous medium leading to the fractional partial integro-differential equation is demonstrated and numerical simulation is established to validate the derived theoretical results.

The general notion of ‘strong’ stability for internal autonomous system descriptions has been recently introduced for continuous and discrete‐time systems. This is a stronger notion of stability compared with alternative definitions (asymptotic, Lyapunov), which prohibits systems described by natural coordinates to have overshooting responses, for arbitrary initial conditions in state space. The paper reviews three refined notions of strong stability, along with the necessary and sufficient conditions corresponding to each notion. Using the Cayley transformation, it is shown that the notions in the two domains are essentially equivalent and that the strong stability conditions can be transformed from one domain to the other in a straightforward way. Copyright © 2013 John Wiley &amp; Sons, Ltd.