Riesz Basis Property of Families of Nonharmonic Exponentials and Application to a Problem of a Radiation of a Vibrating Structure in a Light Fluid

Riesz basis of exponential family for a hyperbolic system

“…It is long this line of thoughts that we try to use spectral information of A to investigate the spectrum-determined growth. The approach that we take to prove our result is different to the one taken in [4,6,8,10,11].…”

“…Since this Riesz basis property is so important, there is extensive literature on this problem; we refer to the works [4,6,9,10] where the same problem is treated for a specific perturbation of a closed linear operator. The basic approach is to estimate eigenvalues, the exponential family of eigenvalues, and then find some transformation to transform the set of the family of a perturbed operator to an orthonormal basis of eigenvectors.…”

Frame of Exponentials Related to Analytic Families Operators and Application to a Non-self Adjoint Problem of Radiation of a Vibrating Structure in a Light Fluid

“…Nevertheless, we find that the Riesz basis property and especially Riesz basis of exponential families attracts the attention of many researchers and have many applications in mathematical physics. Among these applications, we cite the problem of radiation of a vibrating structure in a light fluid initially which was motivated by Filippi et al [18] and widely considered in literature [5,[14][15][16][17]20,21]. Indeed, in [5] the authors studied the existence of a sequence of complex numbers (ε n ) n∈N * such that the family of exponentials associated to the eigenvalues of the operator…”

“…Among these applications, we cite the problem of radiation of a vibrating structure in a light fluid initially which was motivated by Filippi et al [18] and widely considered in literature [5,[14][15][16][17]20,21]. Indeed, in [5] the authors studied the existence of a sequence of complex numbers (ε n ) n∈N * such that the family of exponentials associated to the eigenvalues of the operator…”

“…On the other hand, based on the spectral analysis developed by Nagy [24] and on a stability result of Riesz bases due to Schueller [26], Charfi et al [5] assured, for each eigenvalue λ n of T 0 , the existence of a sequence of complex numbers (ε n ) n∈N * and a sequence of eigenvalues (λ n (ε n )) n∈N * , which can be developed as entire series of (ε n ) n∈N * , such that the system {e iλ n (ε n )t } n∈N * forms a Riesz basis in L 2 (0, T ).…”