In recent investigation 8] concerning the asymptotic behavior of Gram Schmidt or-thonormalization procedure applied to the nonnegative integer shifts of a given function, the problem of determining whether or not such functions form a Riesz system in L 2 (IR +) arose. In this note, we provide a suucient condition to determine whether the nonnegative translates form a Riesz system on L 2 (IR +)....

This paper gives an abstract sufficient condition on Riesz basis with parentheses property for the generators of C0-groups in Hilbert spaces whose eigenvalues are comprised of some finite unification of separable sets after taking the algebraic multiplicities into account. The condition is then applied to the closed-loop system of a serially connected string system under joint damping feedbacks...

In this paper, we consider the boundary stabilization of a sandwich beam which consists of two outer stiff layers and a compliant middle layer. Using Riesz basis approach, we show that there is a sequence of generalized eigenfunctions, which forms a Riesz basis in the state space. As a consequence, the spectrum-determined growth condition as well as the exponential stability of the closed-loop ...

The Riesz basis property of the generalized eigenvector system of a Timoshenko beam with boundary feedback controls applied to two ends is studied in the present paper. The spectral property of the operator A determined by the closed loop system is investigated. It is shown that operator A has compact resolvent and generates a C0 semigroup, and its spectrum consists of two branches and has two ...

(see [9] and [10]). Throughout this work, we assume that the function satis®es the following conditions: (i) j …x†j …cons:† jB…x†j=jxj1‡"; where " 0 and jB…x†j is bounded and 1-periodic function on R. (ii) P n2Z …n† eÿin converges absolutely to a function that has no zeros on ‰ÿ ; Š. It is known that conditions (i) and (ii) imply that fK…x; n†; n 2 Zg is a Riesz basis on V , with a unique biort...

In this paper, we study the generation problem of Riesz basis for a general network of strings with joint damping at each vertex. First, we give a basic spectral property of the system operator A. Under certain conditions, we prove that the spectrum of A is distributed in a strip parallel to the imaginary axis. By the discussion of the completeness of generalized eigenvectors of the operator A,...

Using an abstract condition of Riesz basis generation of discrete operators in the Hilbert spaces, we show, in this paper, that a sequence of generalized eigenfunctions of an Euler– Bernoulli beam equation with a tip mass under boundary linear feedback control forms a Riesz basis for the state Hilbert space. In the meanwhile, an asymptotic expression of eigenvalues and the exponential stability...

In this paper we use the lifting scheme to construct biorthogonal spline wavelet bases on regularly refined non-uniform grids. The wavelets have at least one vanishing moment and on each resolution level they form an L2 Riesz basis. Furthermore we are interested in determining the exact range of Sobolev exponents for which the complete multilevel system forms a Riesz basis. Hereto we need to ex...

A linear feedback control is designed regardless of dissipativity of the system for the stabilization of a flexible beam with a tip rigid body. The Riesz basis approach is adopted in the investigation. It is shown that the closed loop system is a Riesz spectral system and as consequences, the exponential stability, the observability and the controllability of the system are concluded. Finally, ...

We show that there exists a bounded subset of $\mathbb{R}$ such no system exponentials can be Riesz basis for the corresponding Hilbert space. An additional result gives lower bound constant any putative two dimensional disk.