Hilbert Space Of Square-integrable Functions Research Articles

Covariant Integral Quantization of the Semi-Discrete SO(3)-Hypercylinder

Covariant integral quantization with rotational SO(3) symmetry is established for quantum motion on this group manifold. It can also be applied to Gabor signal analysis on this group. The corresponding phase space takes the form of a discrete-continuous hypercylinder. The central tool for implementing this procedure is the Weyl–Gabor operator, a non-unitary operator that operates on the Hilbert space of square-integrable functions on SO(3). This operator serves as the counterpart to the unitary Weyl or displacement operator used in constructing standard Schrödinger–Glauber–Sudarshan coherent states. We unveil a diverse range of properties associated with the quantizations and their corresponding semi-classical phase-space portraits, which are derived from different weight functions on the considered discrete-continuous hypercylinder. Certain classes of these weight functions lead to families of coherent states. Moreover, our approach allows us to define a Wigner distribution, satisfying the standard marginality conditions, along with its related Wigner transform.

INFERENCE ON THE DIMENSION OF THE NONSTATIONARY SUBSPACE IN FUNCTIONAL TIME SERIES

We propose a statistical procedure to determine the dimension of the nonstationary subspace of cointegrated functional time series taking values in the Hilbert space of square-integrable functions defined on a compact interval. The procedure is based on sequential application of a proposed test for the dimension of the nonstationary subspace. To avoid estimation of the long-run covariance operator, our test is based on a variance ratio-type statistic. We derive the asymptotic null distribution and prove consistency of the test. Monte Carlo simulations show good performance of our test and provide evidence that it outperforms the existing testing procedure. We apply our methodology to three empirical examples: age-specific U.S. employment rates, Australian temperature curves, and Ontario electricity demand.

К вопросу о регуляризации классических условий оптимальности в выпуклой задаче оптимального управления c фазовыми ограничениями

We consider the regularization of classical optimality conditions in a convex optimal control problem for a linear system of ordinary differential equations with pointwise state constraints such as equality and inequality, understood as constraints in the Hilbert space of square-integrable functions. The set of admissible task controls is traditionally embedded in the space of square-integrable functions. However, the target functional of the optimization problem is not, generally speaking, strongly convex. Obtaining regularized classical optimality conditions is based on a technique involving the use of two regularization parameters. One of them is used for the regularization of the dual problem, while the other is contained in a strongly convex regularizing addition to the target functional of the original problem. The main purpose of the obtained regularized Lagrange principle and the Pontryagin maximum principle is the stable generation of minimizing approximate solutions in the sense of J. Varga for the purpose of practical solving the considered optimal control problem with pointwise state constraints.

On the Parity Under Metapletic Operators and an Extension of a Result of Lyubarskii and Nes

In this work we show that if the frame property of a Gabor frame with window in Feichtinger’s algebra and a fixed lattice only depends on the parity of the window, then the lattice can be replaced by any other lattice of the same density without losing the frame property. As a byproduct we derive a generalization of a result of Lyubarskii and Nes, who could show that any Gabor system consisting of an odd window function from Feichtinger’s algebra and any separable lattice of density tfrac{n+1}{n}, n in mathbb {N}_+, cannot be a Gabor frame for the Hilbert space of square-integrable functions on the real line. We extend this result by removing the assumption that the lattice has to be separable. This is achieved by exploiting the interplay between the symplectic and the metaplectic group.

Quantum States of a Neutral Massive Fermion with an Anomalous Magnetic Moment in an External Electric Field

The quantum dynamics of a nonrelativistic neutral massive fermion with an anomalous magnetic moment (AMM) is examined in the external electric field of an infinitely long thin homogeneously charged thread in the plane with a normal directed along the thread. The Hamiltonian of the Dirac–Pauli equation for a neutral fermion with AMM is essentially singular in the considered external field and requires a supplementary extension of the definition in order for it to be treated as a self-adjoint quantum-mechanical operator. All one-parameter self-adjoint extensions of the Hamiltonian of the Dirac–Pauli equation in the considered external field are found in the nonrelativistic approximation. The corresponding Hilbert space of squareintegrable functions, including a singularity point of the Hamiltonian, is specified for each self-adjoint extension of the Hamiltonian. The wave functions of free and bound states, as well as discrete energy levels, are determined by the self-adjoint extension method and their correspondence with similar quantities obtained by the physical regularization procedure is discussed. It is shown that energy levels of bound states are simple poles of the scattering amplitude, which should be extended in definition by introducing the self-adjoint extension parameter into it. Expressions for the scattering amplitude and cross-section, depending on the orientation of the initial-state spin of fermion, are obtained.

Quasi-stationary states and fermion pair creation from a vacuum in supercritical Coulomb field

Creation of charged fermion pair from a vacuum in so-called supercritical Coulomb potential is examined for the case when fermions can move only in the same (one) plane. In which case, quantum dynamics of charged massive or massless fermions can be described by the two-dimensional Dirac Hamiltonians with an usual [Formula: see text] Coulomb potential. These Hamiltonians are singular and require the additional definition in order for them to be treated as self-adjoint quantum-mechanical operators. We construct the self-adjoint two-dimensional Dirac Hamiltonians with a Coulomb potential and determine the quantum-mechanical states for such Hamiltonians in the corresponding Hilbert spaces of square-integrable functions. We determine the scattering amplitude in which the self-adjoint extension parameter is incorporated and then obtain equations implicitly defining possible discrete energy spectra of the self-adjoint Dirac Hamiltonians with a Coulomb potential. It is shown that this quantum system becomes unstable in the presence of a supercritical Coulomb potential which manifests in the appearance of quasi-stationary states in the lower (negative) energy continuum. The energy spectrum of those states is quasi-discrete, consists of broadened levels with widths related to the inverse lifetimes of the quasi-stationary states as well as the probability of creation of charged fermion pair by a supercritical Coulomb field. Explicit analytical expressions for the creation probabilities of charged (massive or massless) fermion pair are obtained in a supercritical Coulomb field.

Quantum states of a neutral massive fermion with an anomalous magnetic moment in an Aharonov–Casher field

The planar nonrelativistic quantum dynamics of a neutral massive fermion with an anomalous magnetic moment (AMM) in the electric field of infinitely long and thin thread with a charge density distributed uniformly along it (an Aharonov–Casher field) is examined. The relevant Hamiltonian is singular and requires additional specification of a one-parameter self-adjoint extension, which can be given in terms of physically acceptable boundary conditions. We find all possible self-adjoint Hamiltonians with an Aharonov–Casher field (ACF) by constructing the corresponding Hilbert space of square-integrable functions, including the [Formula: see text] region, for all their Hamiltonians. We determine the most relevant physical quantities, such as energy spectrum and wave functions and discuss their correspondence with those obtained by the physical regularization procedure. We show that energy levels of bound states are simple poles of the scattering amplitude. It is shown that the scattering amplitudes and cross-sections depend essentially on the initial-state spin of fermions.

Completeness of inertial modes of an incompressible inviscid fluid in a corotating ellipsoid.

Inertial modes are the eigenmodes of contained rotating fluids restored by the Coriolis force. When the fluid is incompressible, inviscid, and contained in a rigid container, these modes satisfy Poincaré's equation that has the peculiarity of being hyperbolic with boundary conditions. Inertial modes are, therefore, solutions of an ill-posed boundary-value problem. In this paper, we investigate the mathematical side of this problem. We first show that the Poincaré problem can be formulated in the Hilbert space of square-integrable functions, with no hypothesis on the continuity or the differentiability of velocity fields. We observe that with this formulation, the Poincaré operator is bounded and self-adjoint, and as such, its spectrum is the union of the point spectrum (the set of eigenvalues) and the continuous spectrum only. When the fluid volume is an ellipsoid, we show that the inertial modes form a complete base of polynomial velocity fields for the square-integrable velocity fields defined over the ellipsoid and meeting the boundary conditions. If the ellipsoid is axisymmetric, then the base can be identified with the set of Poincaré modes, first obtained by Bryan [Philos. Trans. R. Soc. London 180, 187 (1889)PTRMAD1364-503X10.1098/rsta.1889.0006], and completed with the geostrophic modes.

Geodesic PCA in the Wasserstein space by convex PCA

We introduce the method of Geodesic Principal Component Analysis (GPCA) on the space of probability measures on the line, with finite second moment, endowed with the Wasserstein metric. We discuss the advantages of this approach, over a standard functional PCA of probability densities in the Hilbert space of square-integrable functions. We establish the consistency of the method by showing that the empirical GPCA converges to its population counterpart, as the sample size tends to infinity. A key property in the study of GPCA is the isometry between the Wasserstein space and a closed convex subset of the space of square-integrable functions, with respect to an appropriate measure. Therefore, we consider the general problem of PCA in a closed convex subset of a separable Hilbert space, which serves as basis for the analysis of GPCA and also has interest in its own right. We provide illustrative examples on simple statistical models, to show the benefits of this approach for data analysis. The method is also applied to a real dataset of population pyramids.

Revisiting Quantum Discord for Two-Qubit X States: The Error Bound to an Analytical Formula

We study the Schr\"odinger operator with a potential given by the sum of the potentials for harmonic oscillator and imaginary cubic oscillator and we focus on its pseudospectral properties. A summary of known results about the operator and its spectrum is provided and the importance of examining its pseudospectrum as well is emphasized. This is achieved by employing scaling techniques and treating the operator using semiclassical methods. The existence of pseudoeigenvalues very far from the spectrum is proven, and as a consequence, the spectrum of the operator is unstable with respect to small perturbations and the operator cannot be similar to a self-adjoint operator via a bounded and boundedly invertible transformation. It is shown that its eigenfunctions form a complete set in the Hilbert space of square-integrable functions; however, they do not form a Schauder basis.

On the Pseudospectrum of the Harmonic Oscillator with Imaginary Cubic Potential

We study the Schrödinger operator with a potential given by the sum of the potentials for harmonic oscillator and imaginary cubic oscillator and we focus on its pseudospectral properties. A summary of known results about the operator and its spectrum is provided and the importance of examining its pseudospectrum as well is emphasized. This is achieved by employing scaling techniques and treating the operator using semiclassical methods. The existence of pseudoeigenvalues very far from the spectrum is proven, and as a consequence, the spectrum of the operator is unstable with respect to small perturbations and the operator cannot be similar to a self-adjoint operator via a bounded and boundedly invertible transformation. It is shown that its eigenfunctions form a complete set in the Hilbert space of square-integrable functions; however, they do not form a Schauder basis.

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.

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.

Lie methods for color robot vision

SUMMARYWe describe how Lie-theoretical methods can be used to analyze color related problems in machine vision. The basic observation is that the nonnegative nature of spectral color signals restricts these functions to be members of a limited, conical section of the larger Hilbert space of square-integrable functions. From this observation, we conclude that the space of color signals can be equipped with a coordinate system consisting of a half-axis and a unit ball with the Lorentz groups as natural transformation group. We introduce the theory of the Lorentz group SU(1, 1) as a natural tool for analyzing color image processing problems and derive some descriptions and algorithms that are useful in the investigation of dynamical color changes. We illustrate the usage of these results by describing how to compress, interpolate, extrapolate, and compensate image sequences generated by dynamical color changes.

Determinants of zeroth order operators

For compact Riemannian manifolds all of whose geodesics are closed (aka Zoll manifolds) one can define the determinant of a zeroth order pseudodifferential operator by mimicking Szego’s definition of this determinant for the operator: multiplication by a bounded function, on the Hilbert space of square-integrable functions on the circle. In this paper we prove that the non-local contribution to this determinant can be computed in terms of a much simpler “zeta-regularized” determinant.

Localized bases in [formula omitted] and their use in the analysis of Brownian motion

Motivated by problems on Brownian motion, we introduce a recursive scheme for a basis construction in the Hilbert space L 2 ( 0 , 1 ) which is analogous to that of Haar and Walsh. More generally, we find a new decomposition theory for the Hilbert space of square-integrable functions on the unit-interval, both with respect to Lebesgue measure, and also with respect to a wider class of self-similar measures μ . That is, we consider recursive and orthogonal decompositions for the Hilbert space L 2 ( μ ) where μ is some self-similar measure on [ 0 , 1 ] . Up to two specific reflection symmetries, our scheme produces infinite families of orthonormal bases in L 2 ( 0 , 1 ) . Our approach is as versatile as the more traditional spline constructions. But while singly generated spline bases typically do not produce orthonormal bases, each of our present algorithms does.

Continuous wavelet transform and wavelet frames on the sphere using Clifford analysis

In this paper we construct a continuous wavelet transform (CWT) on the sphere $S^{n-1}$ based on the conformal group of the sphere, the Lorentz group Spin$(1,n)$. For this purpose, we present a short survey on the existing techniques of continuous wavelet transform and of conformal transformations on the unit sphere. We decompose the conformal group into the maximal compact subgroup of rotations Spin$(n)$ and the set of Mobius transformations of the form $\varphi_a(x) = (x-a)(1+ax)^{-1}$, where $a \in B^n$ and $B^n$ denotes the unit ball in $\mathbb R^n$. Based on a study of the influence of the parameter $a$ arising in the definition of dilations/contractions on the sphere we define a class of local conformal dilation operators and consequently a family of continuous wavelet transforms for the Hilbert space of square integrable functions on the sphere $L_2(S^{n-1})$ and the Hardy space $H^2$. In the end we construct Banach frames for our wavelets and prove Jackson-type theorems for the best $n$-point approximation.

Symplectically covariant Schrödinger equation in phase space

A classical theorem of Stone and von Neumann says that the Schr\"{o}dinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schr\"{o}dinger equation

Efficient estimation of linear functionals of a bivariate distribution with equal, but unknown marginals: the least-squares approach

In this paper, we characterize and construct efficient estimators of linear functionals of a bivariate distribution with equal marginals. An efficient estimator equals the empirical estimator minus a correction term and provides significant improvements over the empirical estimator. We construct an efficient estimator by estimating the correction term. For this we use the least-squares principle and an estimated orthonormal basis for the Hilbert space of square-integrable functions under the unknown equal marginal distribution. Simulations confirm the asymptotic behavior of this estimator in moderate sample sizes and the considerable theoretical gains over the empirical estimator.

ON THE FOUNDATIONS OF GRANULAR COMPUTING PARADIGM

This paper investigates the foundations of granular computing, which is a paradigm oriented towards capturing and processing meaningful pieces of information, the so-called information granules. Each level of granulation induces a higher complexity in the semantic structure of the universe of discourse we have to deal with. Equipping the space of such granules with suitable metric concepts is the key to disclose the cognitive structures underlying granular information systems and to abstract or highlight their hidden meaning. We first interrogate on the epistemological foundations of granular computing and address its hard core: the representation formalism. As far as we are concerning with representing fuzzy granules, the accent is shifted from norms and metrics that equip numeric spaces to those equip-ping function spaces. The Hilbert space of square-integrable functions is the most suitable choice when specific goals are assumed, such as deriving second order statistics (variance or covariance) for random fuzzy-valued variables, or using the projection theorem for solving minimum norm problems. We interrogate on the foundations of univariate and multivariate granular statistics, fuzzy extension of principal component analysis, granular clustering, granular graphical models, granular least squares estimates and discuss some related approaches.