Riesz Basis Research Articles

Initial-Boundary Value Problem for the Beam Vibration Equation in the Multidimensional Case

In the multidimensional case, we study the problem with initial and boundary conditions for the equation of vibrations of a beam with one end clamped and the other hinged. An existence and uniqueness theorem is proved for the posed problem in Sobolev classes. A solution of the problem under consideration is constructed as the sum of a series in the system of eigenfunctions of a multidimensional spectral problem for which the eigenvalues are determined as the roots of a transcendental equation and the system of eigenfunctions is constructed. It is shown that this system of eigenfunctions is complete and forms a Riesz basis in Sobolev spaces. Based on the completeness of the system of eigenfunctions, a theorem about the uniqueness of a solution to the posed initial-boundary value problem is stated.

Characterizations of (near) exact g-frames, g-Riesz bases, and Besselian g-frames

In this paper, we use a new general sequence corresponding to a [Formula: see text]-Bessel sequence [Formula: see text] to characterize that [Formula: see text] are [Formula: see text]-linear independent, [Formula: see text]-complete and a [Formula: see text]-frame. We also use [Formula: see text] and the refinement [Formula: see text] to characterize each other to be (near) exact [Formula: see text]-frames or [Formula: see text]-Riesz bases. Finally, we give several constructions and an equivalent characterization of Besselian [Formula: see text]-frames and near exact [Formula: see text]-frames.

TO THE QUESTION OF A MULTIPOINT MIXED BOUNDARY VALUE PROBLEM FOR A WAVE EQUATION

It is well known that some problems in mechanics and physics lead to partial differential equations of the hyperbolic type. A classical example of the hyperbolic type is wave equation. When posed, the task sometimes lacks the classical boundary condition and the need arises to have a nonlocal boundary condition. Aim our work is get D'Alembert formula for mixed boundary value problem generated by a wave equation. In the classical case, given D'Alembert formula for boundary value problem generated by a wave equation. In our case, we must give D'Alembert formula for mixed boundary value problem. For this, we consider ordinary differential operator L withnon-local boundary conditions. We search the solution of the wave equation like a sum with eigenfunction of the operator L. There are we use that fact, that eigenfunction of the operator L is Riesz basis in L-2 (0, l). Through this method and calculation we get D'Alembert formula.

Completeness Property of One-Dimensional Perturbations of Normal and Spectral Operators Generated by First Order Systems

The paper is concerned with the completeness property of rank one perturbations of the unperturbed operators generated by special boundary value problems (BVP) for the following $$2 \times 2$$ system 0.1 $$\begin{aligned}&L y = -i B^{-1} y' + Q(x) y = \lambda y , \quad B = \begin{pmatrix} b_1 &{}\quad 0 \\ 0 &{}\quad b_2 \end{pmatrix}, \quad y = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}, \end{aligned}$$ on a finite interval assuming that the potential matrix Q is summable, and $$b_1 b_2^{-1} \notin \mathbb {R}$$ (essentially non-Dirac type case). We assume that the unperturbed operator generated by a BVP belongs to one of the following three subclasses of the class of spectral operators: (a) normal operators; (b) operators similar either to a normal or almost normal; (c) operators that meet Riesz basis property with parentheses; We show that in each of the three cases there exists (non-unique) operator generated by a quasi-periodic BVP and its certain rank-one perturbations (in the resolvent sense) generated by special BVPs which are complete while their adjoint are not. In connection with the case (b) we investigate Riesz basis property of quasi-periodic BVP under certain assumptions on the potential matrix Q. We also find a simple formula for the rank of the resolvent difference for operators corresponding to two BVPs for $$n \times n$$ system in terms of the coefficients of linear boundary forms.

On the Riesz basis property of the root functions of a discontinuous boundary problem

In this paper, we consider the nonself‐adjoint discontinuous Sturm Liouville operator with periodic (antiperiodic) boundary condition and compatibility conditions. Asymptotic formulas of eigenvalues and eigenfunctions of the operator are obtained. Using these accurate asymptotic formulas for eigenvalues and eigenfunctions, we prove the basisness of the root functions of the boundary value problem.

Galerkin method with new quadratic spline wavelets for integral and integro-differential equations

The paper is concerned with the wavelet-Galerkin method for the numerical solution of Fredholm linear integral equations and second-order integro-differential equations. We propose a construction of a quadratic spline-wavelet basis on the unit interval, such that the wavelets have three vanishing moments and the shortest support among such wavelets. We prove that this basis is a Riesz basis in the space L 2 0 , 1 . We adapt the basis to homogeneous Dirichlet boundary conditions, and using a tensor product we construct a wavelet basis on the hyperrectangle. We use the wavelet-Galerkin method with the constructed bases for solving integral and integro-differential equations, and we show that the matrices arising from discretization have uniformly bounded condition numbers and that they can be approximated by sparse matrices. We present numerical examples and compare the results with the Galerkin method using other quadratic spline wavelet bases and other methods.

Approximate Duals of $g$-frames and Fusion Frames in Hilbert $C^ast-$modules

In this paper, we study approximate duals of $g$-frames and fusion frames in Hilbert $C^ast-$modules. We get some relations between approximate duals of $g$-frames and biorthogonal Bessel sequences, and using these relations, some results for approximate duals of modular Riesz bases and fusion frames are obtained. Moreover, we generalize the concept of $Q-$approximate duality of $g$-frames and fusion frames to Hilbert $C^ast-$modules, where $Q$ is an adjointable operator, and obtain some properties of this kind of approximate duals.

Riesz basis of exponential family for a hyperbolic system

Abstract This paper studies a linear hyperbolic system with boundary conditions that was first studied under some weaker conditions in [8, 11]. Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. It is shown that the associated linear system is the infinitesimal generator of a C 0 {C_{0}} -semigroup; its spectrum consists of zeros of a sine-type function, and its exponential system { e λ n ⁢ t } n ≥ 1 {\{e^{\lambda_{n}t}\}_{n\geq 1}} constitutes a Riesz basis in L 2 ⁢ [ 0 , T ] {L^{2}[0,T]} . Furthermore, by the spectral analysis method, it is also shown that the linear system has a sequence of eigenvectors, which form a Riesz basis in Hilbert space, and hence the spectrum-determined growth condition is deduced.

NONPARAMETRIC DENSITY ESTIMATION BY B-SPLINE DUALITY

In this article, we propose a new nonparametric density estimator derived from the theory of frames and Riesz bases. In particular, we propose the so-called bi-orthogonal density estimator based on the class of B-splines and derive its theoretical properties, including the asymptotically optimal choice of bandwidth. Detailed theoretical analysis and comparisons of our estimator with existing local basis and kernel density estimators are presented. The estimator is particularly well suited for high-frequency data analysis in financial and economic markets.

Frames, biorthogonal dual and other properties associated with wavelet system on ℝ

It is well known that the system of translates in [Formula: see text] has been characterized as Bessel sequences, frames and Riesz bases in terms of the Fourier transform. The aim of this paper is to obtain similar types of characterization for the wavelet system emerging out of integer translations and dyadic dilations. The existence of the biorthogonal dual system and nonredundancy properties of the wavelet system are also investigated here.

Explicit bounds of complex exponential frames on a complex field

In the context of Riesz basis, studies on classical system of exponentials find their origin in the celebrated 1934’s work of Paley and Wiener. The stability question of exponential frames was considered in 1952 by Duffin and Schaeffer in their seminal paper. This article discusses the stability of complex exponential frames $${\left\{e^{{i\lambda}_{n}t}\right\}_{n\in\mathbb{Z}}}$$ in $${L^{2}(-\pi,\pi)}$$ and presents explicit upper and lower bounds for general complex exponential frames perturbed along the entire complex plane.

The spectral analysis and exponential stability of a bi‐directional coupled wave‐ODE system

In this paper, an unstable linear time invariant (LTI) ODE system is stabilized exponentially by the PDE compensato—a wave equation with Kelvin‐Voigt (K‐V) damping. Direct feedback connections between the ODE system and wave equation are established: The velocity of the wave equation enters the ODE through the variable vt(1,t); meanwhile, the output of the ODE is fluxed into the wave equation. It is found that the spectrum of the system operator is composed of two parts: point spectrum and continuous spectrum. The continuous spectrum consists of an isolated point , and there are two branches of asymptotic eigenvalues: the first branch approaches to , and the other branch tends to −∞. It is shown that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the Hilbert state space. As a consequence, the spectrum‐determined growth condition and exponential stability of the system are concluded.

Skip-free Markov chains

The aim of this paper is to develop a general theory for the class of skip-free Markov chains on denumerable state space. This encompasses their potential theory via an explicit characterization of their potential kernel expressed in terms of family of fundamental excessive functions, which are defined by means of the theory of Martin boundary. We also describe their fluctuation theory generalizing the celebrated fluctuations identities that were obtained by using the Wiener-Hopf factorization for the specific skip-free random walks. We proceed by resorting to the concept of similarity to identify the class of skip-free Markov chains whose transition operator has only real and simple eigenvalues. We manage to find a set of sufficient and easy-to-check conditions on the one-step transition probability for a Markov chain to belong to this class. We also study several properties of this class including their spectral expansions given in terms of Riesz basis, derive a necessary and sufficient condition for this class to exhibit a separation cutoff, and give a tighter bound on its convergence rate to stationarity than existing results.

Three Problems on Exponential Bases

AbstractWe consider three special and significant cases of the following problem. Let $D\subset \mathbb{R}^{d}$ be a (possibly unbounded) set of finite Lebesgue measure. Let $E(\mathbb{Z}^{d})=\{e^{2\unicode[STIX]{x1D70B}ix\cdot n}\}\text{}_{n\in \mathbb{Z}^{d}}$ be the standard exponential basis on the unit cube of $\mathbb{R}^{d}$. Find conditions on $D$ for which $E(\mathbb{Z}^{d})$ is a frame, a Riesz sequence, or a Riesz basis for $L^{2}(D)$.

A Multi-Term Basis Criterion for Families of Dilated Periodic Functions

In this paper we formulate a concrete method for determining whether a system of dilated periodic functions forms a Riesz basis in L^2(0,1) . This method relies on a general framework developed by Hedenmalm, Lindqvist and Seip about 20 years ago, which turns the basis question into one about the localisation of the zeros and poles of a corresponding analytic multiplier. Our results improve upon various criteria formulated previously, which give sufficient conditions for invertibility of the multiplier in terms of sharp estimates on the Fourier coefficients. Our focus is on the concrete verification of the hypotheses by means of analytical or accurate numerical approximations. We then examine the basis question for profiles in a neighbourhood of a non-basis family generated by periodic jump functions. For one of these profiles, the p -sine functions, we determine a threshold for positive answer to the basis question which improves upon those found recently.

Distribution Frames and Bases

In this paper we will consider, in the abstract setting of rigged Hilbert spaces, distribution valued functions and we will investigate, in particular, conditions for them to constitute a "continuous basis" for the smallest space $\mathcal D$ of a rigged Hilbert space. This analysis requires suitable extensions of familiar notions as those of frame, Riesz basis and orthonormal basis. A motivation for this study comes from the Gel'fand-Maurin theorem which states, under certain conditions, the existence of a family of generalized eigenvectors of an essentially self-adjoint operator on a domain $\mathcal D$ which acts like an orthonormal basis of the Hilbert space $\mathcal H$. The corresponding object will be called here a {\em Gel'fand distribution basis}. The main results are obtained in terms of properties of a conveniently defined {\em synthesis operator}.

Nonparametric Density Estimation by B-spline Duality

In this paper, we propose a new non-parametric density estimator derived from the theory of frames and Riesz bases. In particular, we propose the so-called bi-orthogonal density estimator based on the class of B-splines, and derive its theoretical properties including the asymptotically optimal choice of bandwidth. Detailed theoretical analysis and comparisons of our estimator with existing local basis and kernel density estimators are presented. The estimator is particularly well suited for high frequency data analysis in financial and economic markets.

Non-self-adjoint operators with real spectra and extensions of quantum mechanics

In this article, we review the quantum mechanical setting associated with a non-self-adjoint Hamiltonian with a real spectrum. The spectral properties of the Hamiltonian of a Swanson-like model are investigated. The eigenfunctions associated with the real simple eigenvalues are shown to form complete systems but not a (Riesz) basis, which gives rise to difficulties in the rigorous mathematical formulation of quantum mechanics. A new inner product, appropriate for the physical interpretation of the model, has been consistently introduced. The dynamics of the system is described. Some specificities of the theory of non-self-adjoint operators with implications in quantum mechanics are discussed.

Controllability for a string with attached masses and Riesz bases for asymmetric spaces

We consider the problem of boundary control for a vibrating string with $N$ interior point masses. We assume the control of Dirichlet, or Neumann, or mixed type is at the left end, and the string is fixed at the right end. Singularities in waves are 'smoothed' out to one order as they cross a point mass. We characterize the reachable set for an $L^2$ control. The control problem is reduced to a moment problem, which is then solved using the theory of exponential divided differences in tandem with unique shape and velocity controllability results. The results are sharp with respect to both the regularity of the solution and with respect to time. The eigenfunctions of the associated Sturm--Liouville problem are used to construct Riesz bases for a family of asymmetric spaces that include the sets of reachable positions and velocities.

Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities

In this paper, by means of the Riesz basis approach, we study the stability of a weakly damped system of two second order evolution equations coupled through the velocities (see (1.1)). If the fractional order damping becomes viscous and the waves propagate with equal speeds, we prove exponential stability of the system and, otherwise, we establish an optimal polynomial decay rate. Finally, we provide some illustrative examples.