Applications of Affine Systems of Walsh Type to Generate Smooth Basis

The question on affine Riesz basis of Walsh affine systems is considered. An affine Riesz basis is constructed, generated by a continuous periodic function that belongs to the space on the real line, which has a derivative almost everywhere; in connection with the construction of this example, we note that the functions of the classical Walsh system suffer a discontinuity and their derivatives almost vanish everywhere. A method of regularization (improvement of differential properties) of the generating function of Walsh affine system is proposed, and a criterion for an affine Riesz basis for a regularized generating function that can be represented as a sum of a series in the Rademacher system is obtained.

Riesz basis property and exponential stability for one-dimensional thermoelastic system with variable coefficients

In this paper, we study Riesz basis property and stability for a nonuniform thermoelastic system with Dirichlet-Dirichlet boundary condition, where the heat subsystem is considered as a control to the whole coupled system. By means of the matrix operator pencil method, we obtain the asymptotic expressions of the eigenpairs, which are exactly coincident to the constant coefficients case} We then show that there exists a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state space and the spectrum determined growth condition is therefore proved. As a result, the exponential stability of the system is concluded.

Hilbert–Schmidt frames and their duals

The concept of Hilbert–Schmidt frame (HS-frame) was first introduced by Sadeghi and Arefijamaal in 2012. It is more general than [Formula: see text]-frames, and thus, covers many generalizations of frames. This paper addresses the theory of HS-frames. We present a parametric and algebraic formula for all duals of an arbitrarily given HS-frame; prove that the canonical HS-dual induces a minimal-norm expression of the elements in Hilbert spaces; characterize when an HS-frame is an HS-Riesz basis, and when an HS-Bessel sequence is an HS-Riesz sequence (HS-Riesz basis) in terms of Gram matrices.

Hamiltonians Generated by Parseval Frames

It is known that self-adjoint Hamiltonians with purely discrete eigenvalues can be written as (infinite) linear combination of mutually orthogonal projectors with eigenvalues as coefficients of the expansion. The projectors are defined by the eigenvectors of the Hamiltonians. In some recent papers, this expansion has been extended to the case in which these eigenvectors form a Riesz basis or, more recently, a ${\mathcal{D}}$ D -quasi basis (Bagarello and Bellomonte in J. Phys. A 50:145203, 2017, Bagarello et al. in J. Math. Phys. 59:033506, 2018), rather than an orthonormal basis. Here we discuss what can be done when these sets are replaced by Parseval frames. This interest is motivated by physical reasons, and in particular by the fact that the mathematical Hilbert space where the physical system is originally defined, contains sometimes also states which cannot really be occupied by the physical system itself. In particular, we show what changes in the spectrum of the observables, when going from orthonormal bases to Parseval frames. In this perspective we propose the notion of $E$ E -connection for observables. Several examples are discussed.

Boundary null-controllability of two coupled parabolic equations : simultaneous condensation of eigenvalues and eigenfunctions.

Let the matrix operator $L=D\partial_{xx}+q(x)A_0 $, with $D=diag(1,\nu)$, $\nu\neq 1$, $q\in L^{\infty}(0,\pi)$, and $A_0$ is a Jordan block of order $1$. We analyze the boundary null controllability for the system $y_{t}-Ly=0$. When $\sqrt{\nu} \notin \mathbb{Q}_{+}^*$ and $q$ is constant, $q=1$ for instance, there exists a family of root vectors of $(L^*,\mathcal{D}(L^*))$ forming a Riesz basis of $L^{2}(0,\pi;\mathbb{R}^2 )$. Moreover in \cite{JFA14} the authors show the existence of a minimal time of control depending on condensation of eigenvalues of $(L^*,\mathcal{D}(L^*))$, that is to say the existence of $T_0(\nu)$ such that the system is null controllable at time $T > T_0(\nu)$ and not null controllable at time $T < T_0(\nu)$. In the same paper, the authors prove that for all $\tau \in [0, +\infty]$, there exists $\nu \in ]0, +\infty[$ such that $T_0(\nu)=\tau$. When $q$ depends on $x$, the property of Riesz basis is no more guaranteed. This leads to a new phenomena: simultaneous condensation of eigenvalues and eigenfunctions. This condensation affects the time of null controllability.

A nonlocal problem for a differential operator of even order with involution

We study a nonlocal problem for ordinary differential equations of {2n}-order with involution. Spectral properties of the operator of this problem are analyzed and conditions for the existence and uniqueness of its solution are established. It is also proved that the system of eigenfunctions of the analyzed problem forms a Riesz basis.

The Dirichlet Problem for the Perturbed Elliptic Equation

In this paper, the authors consider the construction of one class of perturbed problems to the Dirichlet problem for the elliptic equation. The operators of both problems are isospectral, which makes it possible to construct solutions to the perturbed problem using the Fourier method. This article focuses on the Dirichlet problem for the elliptic equation perturbed by the selected variable. We established the spectral properties of the perturbed operator. In this work, we found the eigenvalues and eigenfunctions of the perturbed task operator. Further, we proved the completeness, minimal spanning system, and Riesz basis system of eigenfunctions of the perturbed operator. Finally, we proved the theorem on the existence and uniqueness of the solution to the boundary value problem for a perturbed elliptic equation.