Space Of Square Integrable Functions Research Articles

The Kerzman–Stein operator for piecewise continuously differentiable regions

The Kerzman–Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on a rectifiable curve. If the curve is continuously differentiable, the Kerzman–Stein operator is compact on the Hilbert space of square integrable functions; when there is a corner, the operator is noncompact. Here, we give a complete description of the spectrum for a finite symmetric wedge and we show how this reveals the essential spectrum for curves that are piecewise continuously differentiable. We also give an explicit construction for a smooth curve whose Kerzman–Stein operator has large norm.

On sufficient conditions and stability of shearlet frame

In 2011, Kittipoom et al. introduced a new class of shearlet generators, and gave many good properties. For example, the new class of shearlet generators is dense in the space of square integrable functions and the shearlet frame generated by a member of the new class of shearlet generators possesses the strong HAP etc. In this paper, our main goal is to study sufficient conditions and the stability of shearlet frame. Specifically, we first adjust the definition of shearlet group introduced by Kittipoom et al. through referring to the definition of shearlet group introduced by Dahlke et al., and make each element of the new class of shearlet generators to be admissible. Secondly, we show that a function from the new class of shearlet generators generates shearlet frames for every sufficiently dense and relatively separated sequence. Finally, we prove that a frame generated by a function from the space remains a frame when the time, scale and shear parameters and the generating function undergo small perturbations. These conclusions make shearlet framework to possess great flexibility and practicality in engineering applications.

On Solvability and Waveform Relaxation Methods of Linear Variable-Coefficient Differential-Algebraic Equations

This paper is concerned with the solvability and waveform relaxation methods of linear variable-coefficient differential-algebraic equations (DAEs). Most of the previous works have been focused on linear variable-coefficient DAEs with smooth coefficients and data, yet no results related to the convergence rate of the corresponding waveform relaxation methods has been obtained. In this paper, we develope the solvability theory for the linear variable-coefficient DAEs on Legesgue square-integrable function space in both traditional and least squares senses, and determine the convergence rate of the waveform relaxation methods for solving linear variable-coefficient DAEs.

Hagedorn Wavepackets in Time-Frequency and Phase Space

The Hermite functions are an orthonormalbasis of the space of square integrable functions with favourable approximation properties. Allowing for a flexible localization in position and momentum, the Hagedorn wavepackets generalize the Hermite functions also to several dimensions. Using Hagedorn’s raising and lowering operators, we derive explicit formulas and recurrence relations for the Wigner and FBI transform of the wavepackets and show their relation to the Laguerre polyomials.

Tight frames, partial isometries, and signal reconstruction

This article gives a procedure to convert a frame which is not a tight frame into a Parseval frame for the same space, with the requirement that each element in the resulting Parseval frame can be explicitly written as a linear combination of the elements in the original frame. Several examples are considered, such as a Fourier frame on a spiral. The procedure can be applied to the construction of Parseval frames for the space of square integrable functions whose domain is the ball of radius When a finite number of measurements is used to reconstruct a signal in error estimates arising from such approximation are discussed.

On the Cauchy problem for the Hartree type equation in the Wiener algebra

We consider the mass-subcritical Hartree equation with a homogeneous kernel in the space of square integrable functions whose Fourier transform is integrable. We prove a global well-posedness result in this space. On the other hand, we show that the Cauchy problem is not even locally well-posed if we simply work in the space of functions whose Fourier transform is integrable. Similar results are proven when the kernel is not homogeneous and is such that its Fourier transform belongs to some Lebesgue space.

Oscillators in a (2+1)-dimensional noncommutative space

We study the Harmonic and Dirac Oscillator problem extended to a three-dimensional noncommutative space where the noncommutativity is induced by the shift of the dynamical variables with generators of \documentclass[12pt]{minimal}\begin{document}$SL(2,\mathbb {R})$\end{document}SL(2,R) in a unitary irreducible representation considered in Falomir et al. [Phys. Rev. D 86, 105035 (2012)]. This redefinition is interpreted in the framework of the Levi's decomposition of the deformed algebra satisfied by the noncommutative variables. The Hilbert space gets the structure of a direct product with the representation space as a factor, where there exist operators which realize the algebra of Lorentz transformations. The spectrum of these models are considered in perturbation theory, both for small and large noncommutativity parameters, finding no constraints between coordinates and momenta noncommutativity parameters. Since the representation space of the unitary irreducible representations \documentclass[12pt]{minimal}\begin{document}$SL(2,\mathbb {R})$\end{document}SL(2,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.

Fractal Functions of Discontinuous Approximation

A procedure for the definition of discontinuous real functions is developed, based on a fractal methodology. For this purpose, a binary operation in the space of bounded functions on an interval is established. Two functions give rise to a new one, called in the paper fractal convolution of the originals, whose graph is discontinuous and has a fractal structure in general. The new function approximates one of the chosen pair and, under certain conditions, is continuous. The convolution is used for the definition of discontinuous bases of the space of square integrable functions, whose elements are as close to a classical orthonormal system as desired.

Expansions of Functions Based on Rational Orthogonal Basis with Nonnegative Instantaneous Frequencies

We consider in this paper expansions of functions based on the rational orthogonal basis for the space of square integrable functions. The basis functions have nonnegative instantaneous frequencies so that the expansions make physical sense. We discuss the almost everywhere convergence of the expansions and develop a fast algorithm for computing the coefficients arising in the expansions by combining the characterization of the coefficients with the fast Fourier transform.

Spectrum of the Kerzman–Stein operator for a family of smooth regions in the plane

The Kerzman–Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on the boundary. For bounded regions with smooth boundary, the Kerzman–Stein operator is compact on the Hilbert space of square integrable functions. Here we give an explicit computation of its Hilbert–Schmidt norm for a family of simply connected regions. We also give an explicit computation of the Cauchy operator acting on an orthonormal basis, and we give estimates for the norms of the Kerzman–Stein and Cauchy operators on these regions. The regions are the first regions that display no apparent Möbius symmetry for which there now is explicit spectral information.

How to spoil a good basis set for Rayleigh-Ritz calculations

For model quantum mechanical systems such as the harmonic oscillator and a particle in an impenetrable box, we consider the set of exact discrete spectrum functions and define the modified basis set by subtraction of the ground state wavefunction from all the other wavefunctions with some real weights. It is demonstrated that the modified set of functions is complete in the space of square integrable functions if and only if the series of the squared weights diverges. A similar, but nonequivalent criterion is derived for convergence of Rayleigh-Ritz ground state energy calculations to the exact ground state energy value with the basis set extension. Some numerical illustrations are provided which demonstrate a wide variety of possible situations for model systems.

Ergodicity of the Action of K* on

Connes gave a spectral interpretation of the critical zeros of zeta- and L-functions for a global field K using a space of square integrable functions on the space A_K/K* of adele classes. It is known that for K=Q the space A_K/K* cannot be understood classically, or in other words, the action of Q* on A_Q is ergodic. We prove that the same is true for any global field K, in both the number field and function field cases.

A projected iterative method based on integral equations for inverse heat conduction in domains with a cut

The Cauchy problem for the parabolic heat equation, consisting of the reconstruction of the solution from knowledge of the temperature and heat flux on a part of the boundary of the solution domain, is investigated in a planar region containing a cut. This linear inverse ill-posed problem is numerically solved using an iterative regularization procedure, where at each iteration step mixed Dirichlet–Neumann problems for the parabolic heat equation are used. Using the method of Rothe these mixed problems are reduced to a sequence of boundary integral equations. The integral equations have a square root singularity in the densities and logarithmic and hypersingularities in the kernels. Moreover, the mixed parabolic problems have singularities near the endpoints of the cut. Special techniques are employed to handle each of these (four) types of singularities, and analysis is performed in weighted spaces of square integrable functions. Numerical examples are included showing that the proposed regularizing procedure gives stable and accurate approximations.

Intrinsic Spectral Analysis

It has long been posited that the space of speech sounds is inherently low dimensional, the result of a relatively small number of degrees of freedom involved in the human vocal apparatus. We attempt to formalize this notion by analyzing a simple physical model of the vocal tract and demonstrating that it produces transfer functions whose spectra are restricted to low dimensional manifolds embedded in an infinite dimensional space of square integrable functions. While source convolution and channel distortion precludes analytic recovery of the articulatory configuration from the observed signal, we present a data-driven unsupervised learning algorithm called Intrinsic Spectral Analysis designed to recover from a stream of unannotated and unsegmented audio a set of nonlinear basis functions for the speech manifold. Projecting a traditional spectrogram onto this nonlinear basis defines a novel acoustic representation that is demonstrated to have phonological significance, improved phonetic separability, inherent speaker independence, and complementarity with standard acoustic front-ends.

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.