Abstract
In this paper, we are mainly concerned with a one-dimensional wave control system. We assert the existence of non-orthogonal fusion frames by extending this problem to a theoretical one introduced by Sz. Nagy (Acta Sci Math Szeged 14, 1951). The key idea of this work is based on the estimate inspired from Sz. Nagy (1951) using the spectral analysis method. More precisely, we prove that if the eigenvalues of the unperturbed operator are isolated and with finite multiplicity, we can construct non-orthogonal fusion frames.
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References
Asgari, M.S.: New characterizations of fusion frames \((\)frames of subspaces\()\). Proc. Indian Acad. Sci. Math. Sci. 119, 369–382 (2009)
Asgari, M.S., Khosravi, A.: Frames and bases of subspaces in Hilbert spaces. J. Math. Anal. Appl. 308, 541–553 (2005)
Ben Ali, N., Jeribi, A.: On the Riesz basis of a family of analytic operators in the sense of Kato and application to the problem of radiation of a vibrating structure in a light fluid. J. Math. Anal. Appl. 320, 78–94 (2006)
Cahill, J., Casazza, P.G., Li, S.: Non-orthogonal fusion frames and the sparsity of fusion frame operators. J. Fourier Anal. Appl. 18, 287–308 (2012)
Casazza, P.G., Kutyniok, G.: Frames of Subspaces. Wavelets, Frames and Operator Theory, Contemporary Mathematics, Volume 345, pp. 87–113. American Mathematical Society, Providence (2004)
Christensen, O.: Frames containing a Riesz basis and approximation of the frame coefficients using finite-dimensional methods. J. Math. Anal. Appl. 199, 256–270 (1996)
Ellouz, H., Feki, I., Jeribi, A.: On a Riesz basis of exponentials related to the eigenvalues of an analytic operator and application to a non-selfadjoint problem deduced from a perturbation method for sound radiation. J. Math. Phys. 54, 112101 (2013)
Feki, I., Jeribi, A., Sfaxi, R.: On an unconditional basis of generalized eigenvectors of an analytic operator and application to a problem of radiation of a vibrating structure in a light fluid. J. Math. Anal. Appl. 375, 261–269 (2011)
Feki, I., Jeribi, A., Sfaxi, R.: On a Schauder basis related to the eigenvectors of a family of non-selfadjoint analytic operators and applications. Anal. Math. Phys. 3, 311–331 (2013)
Feki, I., Jeribi, A., Sfaxi, R.: On a Riesz basis of eigenvectors of a nonself-adjoint analytic operator and applications. Linear Multilinear Algebra 62, 1049–1068 (2014)
Guo, B.Z., Chan, K.Y.: Riesz basis generation, eigenvalues distribution, and exponential stability for a Euler–Bernoulli beam with joint feedback control. Rev. Mat. Complut. 14, 205–229 (2001)
Jeribi, A.: Spectral Theory and Applications of Linear Operators and Block Operator Matrices. Springer, New York (2015)
Jeribi, A.: Denseness. Bases and Frames in Banach Spaces and Applications. De Gruyter, Berlin (2018)
Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980)
Kobayashi, T.: Stabilization of infinite-dimensional undamped second order systems by using a parallel compensator. IMA J. Math. Control Inf. 21, 85–94 (2004)
Sz. Nagy, B.: Perturbations des transformations linéaires fermées. Acta Sci. Math. Szeged 14, 125–137 (1951)
Xu, G.Q.: Stabilization of string system with linear boundary feedback. Nonlinear Anal. Hybrid Syst. 1, 383–397 (2007)
Xu, G.Q., Feng, D.X.: The Riesz basis property of a Timoshenko beam with boundary feedback and application. IMA J. Appl. Math. 67, 357–370 (2002)
Xu, G.Q., Yung, S.P.: Properties of a class of \(C_0\) semigroups on Banach spaces and their applications. J. Math. Anal. Appl. 328, 245–256 (2007)
Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, London (1980)
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Ellouz, H., Feki, I. & Jeribi, A. Non-orthogonal Fusion Frames of an Analytic Operator and Application to a One-Dimensional Wave Control System. Mediterr. J. Math. 16, 52 (2019). https://doi.org/10.1007/s00009-019-1318-x
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DOI: https://doi.org/10.1007/s00009-019-1318-x