Space Of Square Integrable Functions Research Articles

On sparse representation of analytic signal in Hardy space

This paper is concerned with the sparse representation of analytic signal in Hardy space , where is the open unit disk in the complex plane. In recent years, adaptive Fourier decomposition has attracted considerable attention in the area of signal analysis in . As a continuation of adaptive Fourier decomposition‐related studies, this paper proves rapid decay properties of singular values of the dictionary. The rapid decay properties lay a foundation for applications of compressed sensing based on this dictionary. Through Hardy space decomposition, this program contributes to sparse representations of signals in the most commonly used function spaces, namely, the spaces of square integrable functions in various contexts. Numerical examples are given in which both compressed sensing and ℓ1‐minimization are used. Copyright © 2013 John Wiley & Sons, Ltd.

On the Bound States for the Three-Body Schrödinger Equation with Decaying Potentials

The three-body Schrodinger operator in the space of square integrable functions is found to be a certain extension of operators which generate the exponential unitary group containing a subgroup with nilpotent Lie algebra of length \({\kappa + 1, \kappa = 0, 1, \ldots}\) As a result, the solutions to the three-body Schrodinger equation with decaying potentials are shown to exist in the commutator subalgebras. For the Coulomb three-body system, it turns out that the task is to solve—in these subalgebras—the radial Schrodinger equation in three dimensions with the inverse power potential of the form \({r^{-{\kappa}-1}}\) . As an application to Coulombic system, analytic solutions for some lower bound states are presented. Under conditions pertinent to the three-unit-charge system, obtained solutions, with \({\kappa = 0}\) , are reduced to the well-known eigenvalues of bound states at threshold.

A Time-Frequency Density Criterion for Operator Identification

We establish a necessary density criterion for the identifiability of time-frequency structured classes of Hilbert-Schmidt operators. The density condition is based on the density criterion for Gabor frames and Riesz bases in the space of square integrable functions. We complement our findings with examples of identifiable operator classes.

High-frequency averaging in the semi-classical singular Hartree equation

We study the asymptotic behavior of the Schrodinger equation in the presence of a nonlinearity of Hartree type in the semi-classical regime. Our scaling corresponds to a weakly nonlinear regime where the nonlinearity affects the leading order amplitude of the solution without altering the rapid oscillations. We show the validity of the WKB-analysis when the potential in the nonlinearity is singular around the origin. No new resonant wave is created in our model, this phenomenon is inhibited due to the nonlinearity. The nonlocal nature of this latter leads us to build our result on a high-frequency averaging effects. In the proof we make use of the Wiener algebra and the space of square-integrable functions.

Noncommutativity in (2+1)-dimensions and the Lorentz group

In this article we considered models of particles living in a three-dimensional space-time with a nonstandard noncommutativity induced by shifting canonical coordinates and momenta with generators of a unitary irreducible representation of the Lorentz group. The Hilbert space gets the structure of a direct product with the representation space, where we are able to construct operators which realize the algebra of Lorentz transformations. We study the modified Landau problem for both Schr\"odinger and Dirac particles, whose Hamiltonians are obtained through a kind of non-Abelian Bopp's shift of the dynamical variables from the ones of the usual problem in the normal space. The spectrum of these models are considered in perturbation theory, both for small and large noncommutativity parameters. We find no constraint between the parameters referring to noncommutativity in coordinates and momenta but they rather play similar roles. Since the representation space of the unitary irreducible representations $SL(2,\mathbb{R})$ can be realized in terms of spaces of square-integrable functions, we conclude that these models are equivalent to quantum mechanical models of particles living in a space with an additional compact dimension.

Analysis of the Volume-Constrained Peridynamic Navier Equation of Linear Elasticity

Well-posedness results for the state-based peridynamic nonlocal continuum model of solid mechanics are established with the help of a nonlocal vector calculus. The peridynamic strain energy density for an elastic constitutively linear anisotropic heterogeneous solid is expressed in terms of the field operators of that calculus, after which a variational principle for the equilibrium state is defined. The peridynamic Navier equilibrium equation is then derived as the first-order necessary conditions and are shown to reduce, for the case of homogeneous materials, to the classical Navier equation as the extent of nonlocal interactions vanishes. Then, for certain peridynamic constitutive relations, the peridynamic energy space is shown to be equivalent to the space of square-integrable functions; this result leads to well-posedness results for volume-constrained problems of both the Dirichlet and Neumann types. Using standard results, well-posedness is also established for the time-dependent peridynamic equation of motion.

The Domain of the Fourier Integral

We consider the problem of determining the Fourier integral in the Hilbert space of square integrable func- tions. Fourier integral is the scalar product of two functions belonging to the Hilbert space of square integrable functions and the Hilbert space of almost periodic functions. Scalar product for different Hilbert spaces defined at the intersection of these spaces, which contains only one zero element. Therefore, the Fourier integral is not defined in the Hilbert space of square integrable functions with nonzero norm.

An orthogonal system of monogenic polynomials over prolate spheroids in [formula omitted

The object of this paper is to construct a complete orthogonal system of monogenic polynomials as solutions of the Moisil–Théodoresco system over prolate spheroids in R3. This will be done in the spaces of square integrable functions over H. A big breakthrough is that the orthogonality of the polynomials in question does not depend on the shape of the spheroids, but only on the location of the foci of the ellipse generating the spheroid. The representations of these polynomials are given explicitly, ready to be implemented on a computer. In addition, we show a corresponding orthogonality of the same polynomials over the surface of the spheroids with respect to a suitable weight function.

Multiresolution Analysis for Markov Interval Maps

We set up a multiresolution analysis on fractal sets derived from limit sets of Markov Interval Maps. For this we consider the ℤ-convolution of a nonatomic measure supported on the limit set of such systems and give a thorough investigation of the space of square integrable functions with respect to this measure. We define an abstract multiresolution analysis, prove the existence of mother wavelets, and then apply these abstract results to Markov Interval Maps. Even though, in our setting the corresponding scaling operators are in general not unitary we are able to give a complete description of the multiresolution analysis in terms of multiwavelets.

Unitary representations of a loop [formula omitted] group, Wiener measure and Γ-function

We construct a family of irreducible unitary representations of the loop affine group of a line (ax+b group) with central extension on the Hilbert space of square-integrable functions with respect to the Wiener measure. We relate the matrix coefficients of the elements of the loop ax+b group to the loop analogue of the Γ-function.

Solution of plane Dirichlet problem using compactly supported 2D wavelet scaling functions

The paper presents the solution of the homogeneous plane Dirichlet problem using the wavelet-Galerkin method with various 2D compactly supported wavelet scaling functions. An analysis of approximation accuracy was performed with respect to the orders of investigated wavelet scaling functions and the level of approximation. The most effective scaling functions for solving the Dirichlet problem were indicated and discussed. The wavelets theory is relatively new, but an effective mathematical tool for solving many theoretical and engineering problems. The wide range of its applica� tions covers developed numerical algorithms for approximation of ordinary differ� ential equations (ODE) and partial differential equations (PDE), image processing, signal analysis etc. The algorithms based on wavelet theory for solving ODE and PDE offer several advantages in comparison with the algorithms based on finite difference and finite element methods, e.g. low timeconsumption and good accu� racy (1). The two most popular waveletbased methods for solvi ng differential equations (DE) are (2): the collocation method and the waveletGalerkin method (WGM). According to some good properties, e.g. the possibility of multiresolution analysis and stability of decomposition and reconstruction in spaces of squareintegrable functions L 2 (R n ), solution stability, WGM gain great popularity. The crucial task in application of waveletbased num erical approximation algo� rithms is the choice of appropriate wavelet scaling functions. In the function ap� proximation problems, the wavelets with infinite support results in poor effects, therefore the orthogonal wavelets were primarily chosen for solving DE (3). One of the most popular wavelets is the Daubechies' ( db ) wavelet, which is often used for solving DE (4, 5). However, several authors used other wavelet scaling func� tions for solving DE, e.g. symlets ( sym ), coiflets ( coif ), Bspline wavelets ( bsp ), biorthogonal ( bior ) and reversed biorthogonal ( rbio ) wavelets (6�8). Due to the separability property of the abovementio ned compactly supported wavelets, they could be simply generalized to the higher dimensions by tensor

Feynman Formulas Representation of Semigroups Generated by Parabolic Difference-Differential Equations

We establish that the Laplas operator with perturbation by symmetrised linear hall of displacement argument operators is the generator of unitary group in the Hilbert space of square integrable functions. The representation of semigroup of Cauchy problem solutions for considered functional differential equation is given by the Feynman formulas.

Spectral approach for kernel-based interpolation

We describe how the resolution of a kernel-based interpolation problem can be associated with a spectral problem. An integral operator is defined from the embedding of the considered Hilbert subspace into an auxiliary Hilbert space of square-integrable functions. We finally obtain a spectral representation of the interpolating elements which allows their approximation by spectral truncation. As an illustration, we show how this approach can be used to enforce boundary conditions in kernel-based interpolation models and in what it offers an interesting alternative for dimension reduction.

SOLUTION OF TIME-PERIODIC WAVE EQUATION USING MIXED FINITE ELEMENTS AND CONTROLLABILITY TECHNIQUES

We consider a controllability method for the time-periodic solution of the two-dimensional scalar wave equation with a first order absorbing boundary condition describing the scattering of a time-harmonic incident wave by a sound-soft obstacle. Solution of the time-harmonic equation is equivalent to finding a periodic solution for the corresponding time-dependent wave equation. We formulate the problem as an exact controllability one and solve the wave equation in time-domain. In a mixed formulation we look for solutions u = (v, p)T. The use of mixed formulation allows us to set the related controllability problem in (L2(Ω))d+1, a space of square-integrable functions in dimension d + 1. No preconditioning is needed when solving this with conjugate gradient method. We present numerical results concerning performance and convergence properties of the method.

Functional regression of continuous state distributions

In this paper, we consider a regression model to study the distributional relationship between economic variables. Unlike the classical regression dealing exclusively with mean relationship, our model can be used to analyze the entire dependent structure in distribution. Technically, we treat density functions as random elements and represent the regression relationship as a compact linear operator in the Hilbert spaces of square integrable functions. We propose a consistent estimation procedure for our model, and develop a test to investigate the dependent structure of moments. An empirical example is provided to illustrate how our methodology can be implemented in practical applications.

On some constructive aspects of monogenic function theory in ℝ4

As is well known, a possible generalization to ℝ4 of the classical Cauchy–Riemann system leads to the so-called Riesz system. The main goal of this paper is to construct explicitly a complete orthonormal system of polynomial solutions of this system with respect to a certain inner product. This will be done in the spaces of square integrable functions on the unit ball over ℝ. Copyright © 2011 John Wiley & Sons, Ltd.

Coherent state transforms attached to generalized Bargmann spaces on the complex plane

AbstractWe construct a family of coherent states transforms attached to generalized Bargmann spaces [C. R. Acad. Sci. Paris, t. 325, 1997] in the complex plane. This constitutes another way of obtaining the kernel of an isometric operator linking the space of square integrable functions on the real line with the true‐poly‐Fock spaces [Oper. Theory, Adv. Appl. v. 117, 2000]. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Classical Mechanics in Hilbert Space, Part 2

We continue from Part 1. We will illustrate the general theory of Hamiltonian mechanics in the Lie group formalism. We then obtain the Hamiltonian formalism in the Hilbert spaces of square integrable functions on the symplectic spaces. We illustrate this general theory with several concrete examples, two of which are the representations of the Lorentz group and the Poincare group with interactions.

Hilbert space for quantum mechanics on superspace

In superspace a realization of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}_2$\end{document}sl2 is generated by the super Laplace operator and the generalized norm squared. In this paper, an inner product on superspace for which this representation is skew-symmetric is considered. This inner product was already defined for spaces of weighted polynomials (see [K. Coulembier, H. De Bie, and F. Sommen, Orthogonality of Hermite polynomials in superspace and Mehler type formulae, Proc. London Math. Soc. (accepted) arXiv:1002.1118]). In this article, it is proven that this inner product can be extended to the super Schwartz space, but not to the space of square integrable functions. Subsequently, the correct Hilbert space corresponding to this inner product is defined and studied. A complete basis of eigenfunctions for general orthosymplectically invariant quantum problems is constructed for this Hilbert space. Then the integrability of the \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}_2$\end{document}sl2-representation is proven. Finally, the Heisenberg uncertainty principle for the super Fourier transform is constructed.

Classical Mechanics in Hilbert Space, Part 1

We consider the Hamilton formulation as well as the Hamiltonian flows on a symplectic (phase) space. These symplectic spaces are derivable from the Lie group of symmetries of the physical system considered. In Part 2 of this work, we then obtain the Hamiltonian formalism in the Hilbert spaces of square integrable functions on the symplectic spaces so obtained.