Weighted Hilbert Space Research Articles

Exponential Convergence-(t,s)-Weak Tractability of Approximation in Weighted Hilbert Spaces

We study L2-approximation problems in the weighted Hilbert spaces in the worst case setting. Three interesting weighted Hilbert spaces appear in this paper, whose weights are equipped with two positive parameters γj and αj for j∈N. We consider algorithms using the class of arbitrary linear functionals. We discuss the exponential convergence-(t,s)-weak tractability of these L2-approximation problems. In particular, we obtain the sufficient and necessary conditions on the weights for exponential convergence-weak tractability and exponential convergence-(t,1)-weak tractability with t<1.

Tractability of Approximation of Functions Defined over Weighted Hilbert Spaces

We investigate L2-approximation problems in the worst case setting in the weighted Hilbert spaces H(KRd,α,γ) with weights Rd,α,γ under parameters 1≥γ1≥γ2≥⋯≥0 and 1<α1≤α2≤⋯. Several interesting weighted Hilbert spaces H(KRd,α,γ) appear in this paper. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. We discuss tractability of L2-approximation problems for the involved Hilbert spaces, which describes how the information complexity depends on d and ε−1. As a consequence we study the strongly polynomial tractability (SPT), polynomial tractability (PT), weak tractability (WT), and (t1,t2)-weak tractability ((t1,t2)-WT) for all t1>1 and t2>0 in terms of the introduced weights under the absolute error criterion or the normalized error criterion.

Separation for Schrodinger-type operators in weighted Hilbert spaces

Separation for Schrodinger-type operators in weighted Hilbert spaces

Data-driven probability density forecast for stochastic dynamical systems

In this paper, a data-driven non-parametric approach is presented for forecasting the probability density evolution of stochastic dynamical systems. The method is based on stochastic Koopman operator and extended dynamic mode decomposition (EDMD). To approximate the finite-dimensional eigendecomposition of the stochastic Koopman operator, EDMD is applied to the training data set sampled from the stationary distribution of the underlying stochastic dynamical system. The family of the Koopman operators form a semigroup, which is generated by the infinitesimal generator of the stochastic dynamical system. A significant connection between the generator and Fokker-Planck operator is leveraged to construct an orthonormal basis of a weighted Hilbert space, enabling a spectral decomposition of the probability density function to be performed in this space. This approach is a data-driven method and used to predict the probability density evolution and real-time moment estimation. In the limit of the large number of snapshots and observables, the data-driven probability density approximation converges to the Galerkin projection of the semigroup solution of Fokker-Planck equation on a basis adapted to an invariant measure. This method shares similarities with diffusion forecasting but is capable of producing more accurate probability density forecasts than diffusion forecasting. A few numerical examples are presented to illustrate the performance of the proposed data-driven probability density forecast.

Elliptic finite-band potentials of a non-self-adjoint Dirac operator

We present an explicit two-parameter family of finite-band Jacobi elliptic potentials given by q≡Adn(x;m), where m∈(0,1) and A can be taken to be positive without loss of generality, for a non-self-adjoint Dirac operator L, which connects two well-known limiting cases of the plane wave (m=0) and of the sech potential (m=1). We show that, if A∈N, then the spectrum consists of R plus 2A Schwarz symmetric segments (bands) on iR. This characterization of the spectrum is obtained by relating the periodic and antiperiodic eigenvalue problems for the Dirac operator to corresponding eigenvalue problems for tridiagonal operators acting on Fourier coefficients in a weighted Hilbert space, and to appropriate connection problems for Heun's equation. Conversely, if A∉N, then the spectrum of L consists of infinitely many bands in C. When A∈N, the corresponding potentials generate finite-genus solutions for all the positive and negative flows associated with the focusing nonlinear Schrödinger hierarchy, including the modified Korteweg-deVries equation and the sine-Gordon equation.

Composition operators on weighted Hilbert spaces of Dirichlet series

AbstractWe study composition operators of characteristic zero on weighted Hilbert spaces of Dirichlet series. For this purpose, we demonstrate the existence of weighted mean counting functions associated with the Dirichlet series symbol, and provide a corresponding change of variables formula for the composition operator. This leads to natural necessary conditions for the boundedness and compactness. For Bergman‐type spaces, we are able to show that the compactness condition is also sufficient, by employing a Schwarz‐type lemma for Dirichlet series.

TE-Polarized Electromagnetic Wave Diffraction by a Circular Slotted Cylinder

The problem of diffraction of a TE-polarized electromagnetic wave by a circular slotted cylinder is investigated. The boundary value problem in question for the Helmholtz equation is reduced to an infinite system of linear algebraic equations of the second kind (SLAE-II) using integral summation identities (ISI). A detailed study of the matrix operator of the problem is performed and its Fredholm property in the weighted Hilbert space of infinite sequences is proven. The convergence of the truncation method constructed in the paper for the numerical solution of SLAE-II is justified and the results of computations are presented and discussed, specifically considering the determination of resonance modes.

Dyson maps and unitary evolution for Maxwell equations in tensor dielectric media

The propagation and scattering of electromagnetic waves in dielectric media is of theoretical and experimental interest in a wide variety of fields. An understanding of observational results generally requires a numerical solution of Maxwell equations---usually implemented on conventional computers using sophisticated numerical algorithms. In recent years, advances in quantum information science and in the development of quantum computers have piqued curiosity about taking advantage of these resources for an alternate numerical approach to Maxwell equations. This requires a reformulation of the classical Maxwell equations into a form suitable for quantum computers which, unlike conventional computers, are limited to unitary operations. In this paper, a unitary framework is developed for the propagation of electromagnetic waves in a spatially inhomogeneous, passive, nondispersive, and anisotropic dielectric medium. For such a medium, generally, the evolution operator in the combined Faraday-Ampere equations is not unitary. There are two steps needed to convert this equation into a unitary evolution equation. In the first step, a weighted Hilbert space is formulated in which the generator of dynamics is a pseudo-Hermitian operator. In the second step, a Dyson map is constructed which maps the weighted-physical-Hilbert space to the original Hilbert space. The resulting evolution equation for the electromagnetic wave fields is unitary. Utilizing the framework developed in these steps, a unitary evolution equation is derived for electromagnetic wave propagation in a uniaxial dielectric medium. The resulting form is suitable for quantum computing.

Dynamics of non-autonomous first order lattice systems in weighted spaces

In a weighted Hilbert space, the existence of a uniform global attractor for a family of first order non-autonomous deterministic lattice dynamical systems with a nonlinear part of the form ft,u=fit,ui−m,…,ui+mi∈Z is carefully investigated, where ft,⋅ is an element of the hull of an almost periodic function in a suitable Banach space.

On a semilinear parabolic problem with non-local (Bitsadze–Samarskii type) boundary conditions in more dimensions

This paper studies a semilinear parabolic equation in a bounded domain Ω⊂Rd along with nonlocal boundary conditions. The boundary values are linked to the values of a solution on an interior (d−1)-dimensional manifold lying inside Ω. Firstly, the solvability of a steady-state problem is addressed. Secondly, involving the semi-discretization in time, a constructive algorithm for approximation of a solution to a transient problem is developed. The well-posedness of the problem in a weighted Hilbert space is shown. Convergence of approximations is addressed and the error estimated are derived. Numerical experiments support the theoretical algorithms.

Weighted composition operators on weighted Bergman and Dirichlet spaces

AbstractWe study the boundedness and compactness of weighted composition operators acting on weighted Bergman spaces and weighted Dirichlet spaces by using the corresponding Carleson measures. We give an estimate for the norm and the essential norm of weighted composition operators between weighted Bergman spaces as well as the composition operators between weighted Hilbert spaces.

On invertibility of some classes of operators in weighted Hilbert spaces

On invertibility of some classes of operators in weighted Hilbert spaces

On the Spectra of General Ordinary Quasi-Differential Operators and Their <i>L</i><sup>2</sup><sub><i>w</i></sub>-Solutions

In this paper, we consider the general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ+ on the interval [a,b). We shall show in the case of one singular end-point and under suitable conditions that all solutions of a general ordinary quasi-differential equation are in the weighted Hilbert space provided that all solutions of the equations and its adjoint are in . Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions may be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while the others are new.

Maximally Accretive Differential Operators of First Order in the Weighted Hilbert Spaces

Maximally Accretive Differential Operators of First Order in the Weighted Hilbert Spaces

A right inverse of Cauchy–Riemann operator ∂ ¯ k +a in the weighted Hilbert space L 2 (ℂ,e -|z| 2 )

Using Hörmander L 2 method for Cauchy–Riemann equations from complex analysis, we study a simple differential operator ∂ ¯ k +a of any order (densely defined and closed) in the weighted Hilbert space L 2 (ℂ,e -|z| 2 ) and prove the existence of a right inverse that is bounded.

Spectral expansions of non-self-adjoint generalized Laguerre semigroups

We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a subset of negative definite functions and we name it the class of generalized Laguerre semigroups. Our approach, which goes beyond the framework of perturbation theory, is based on an in-depth and original analysis of an intertwining relation that we establish between this class and a self-adjoint Markov semigroup, whose spectral expansion is expressed in terms of the classical Laguerre polynomials. As a by-product, we derive smoothness properties for the solution to the associated Cauchy problem as well as for the heat kernel. Our methodology also reveals a variety of possible decays, including the hypocoercivity type phenomena, for the speed of convergence to equilibrium for this class and enables us to provide an interpretation of these in terms of the rate of growth of the weighted Hilbert space norms of the spectral projections. Depending on the analytic properties of the aforementioned negative definite functions, we are led to implement several strategies, which require new developments in a variety of contexts, to derive precise upper bounds for these norms.

Global regularity of weak solutions to the generalized Leray equations and its applications

We investigate a regularity for weak solutions of the following generalized Leray equations \begin{equation*} (-\Delta)^{\alpha}V- \frac{2\alpha-1}{2\alpha}V+V\cdot\nabla V-\frac{1}{2\alpha}x\cdot \nabla V+\nabla P=0, \end{equation*} which arises from the study of self-similar solutions to the generalized Naiver-Stokes equations in $\mathbb R^3$. Firstly, by making use of the vanishing viscosity and developing non-local effects of the fractional diffusion operator, we prove uniform estimates for weak solutions $V$ in the weighted Hilbert space $H^\alpha_{\omega}(\mathbb R^3)$. Via the differences characterization of Besov spaces and the bootstrap argument, we improve the regularity for weak solution from $H^\alpha_{\omega}(\mathbb R^3)$ to $H_{\omega}^{1+\alpha}(\mathbb R^3)$. This regularity result, together linear theory for the non-local Stokes system, lead to pointwise estimates of $V$ which allow us to obtain a natural pointwise property of the self-similar solution constructed in \cite{LXZ}. In particular, we obtain an optimal decay estimate of the self-similar solution to the classical Naiver-Stokes equations by means of the special structure of Oseen tensor. This answers the question proposed by Tsai \cite[Comm. Math. Phys., 328 (2014), 29-44]{T}.

Maximally Dissipative Differential Operators of First Order in the Weighted Hilbert Space

Maximally Dissipative Differential Operators of First Order in the Weighted Hilbert Space

Self-Adjointness in Klein-Gordon Theory on Globally Hyperbolic Spacetimes

We prove essential self-adjointness of the spatial part of the linear Klein-Gordon operator with external potential for a large class of globally hyperbolic manifolds. The proof is conducted by a fusion of new results concerning globally hyperbolic manifolds, the theory of weighted Hilbert spaces and related functional analytic advances.

On the Poincaré-Lelong equation in ℂn

ABSTRACT In this paper, we prove the existence of (global) solutions of the Poincaré-Lelong equation , where f is a d-closed form and is in the weighted Hilbert space with Gaussian measure, i.e. . The novelty of this paper is to apply a weighted version of the Poincaré Lemma for 2-forms and then apply Hörmander's solutions for Cauchy–Riemann equations. In both cases, the same weight is used.