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

Kalenyuk Petro I., Baranetskij Yaroslav O., Kolyasa Lubov I.

Journal of Applied Analysis

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2020

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Vol. 26, nr 2

297--307

EN

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.

2

Riesz basis of exponential family for a hyperbolic system

Intissar Abdelkader, Jeribi Aref, Walha Ines

Journal of Applied Analysis

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2019

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Vol. 25, nr 1

13--23

EN

This paper studies a linear hyperbolic system with boundary conditions thatwas first studied under someweaker 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 C0-semigroup; its spectrum consists of zeros of a sine-type function, and its exponential system {eλnt}n≥1 constitutes a Riesz basis in L2[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.

3

Exact boundary controllability of coupled hyperbolic equations

Avdonin S., Choque Rivero A., de Teresa L.

International Journal of Applied Mathematics and Computer Science

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2013

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Vol. 23, no. 4

701--710

EN

We study the exact boundary controllability of two coupled one dimensional wave equations with a control acting only in one equation. The problem is transformed into a moment problem. This framework has been used in control theory of distributed parameter systems since the classical works of A.G. Butkovsky, H.O. Fattorini and D.L. Russell in the late 1960s to the early 1970s. We use recent results on the Riesz basis property of exponential divided differences.

4

Construction of sampling and interpolating sequences for multi-band signals. The two-band case

Avdonin S., Bulanova A., Moran W.

International Journal of Applied Mathematics and Computer Science

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2007

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Vol. 17, no 2

143-156

EN

Recently several papers have related the production of sampling and interpolating sequences for multi-band signals to the solution of certain kinds of Wiener-Hopf equations. Our approach is based on connections between exponential Riesz bases and the controllability of distributed parameter systems. For the case of two-band signals we derive an operator whose invertibility is equivalent to the existence of a sampling and interpolating sequence, and prove the invertibility of this operator.

5

Ingham-Type Inequalities and Riesz Bases of Divided Differences

Avdonin S., Moran W.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 4

803-820

EN

We study linear combinations of exponentials e^{i lambda_n t}, lambda_n in Lambda in the case where the distance between some points lambda_n tends to zero. We suppose that the sequence Lambda is a finite union of uniformly discrete sequences. In (Avdonin and Ivanov, 2001), necessary and sufficient conditions were given for the family of divided differences of exponentials to form a Riesz basis in space L^2 (0,T). Here we prove that if the upper uniform density of Lambda is less than T/(2 pi), the family of divided differences can be extended to a Riesz basis in L^2 (0,T) by adjoining to {e^{i lambda_n t} } a suitable collection of exponentials. Likewise, if the lower uniform density is greater than T/(2 pi), the family of divided differences can be made into a Riesz basis by removing from {e^{i lambda_n t} } a suitable collection of functions e^{i lambda_n t}. Applications of these results to problems of simultaneous control of elastic strings and beams are given.