Opuscula Math. 37, no. 1 (2017), 141-165
The basis property of eigenfunctions in the problem of a nonhomogeneous damped string
Abstract. The equation which describes the small vibrations of a nonhomogeneous damped string can be rewritten as an abstract Cauchy problem for the densely defined closed operator \(i A\). We prove that the set of root vectors of the operator \(A\) forms a basis of subspaces in a certain Hilbert space \(H\). Furthermore, we give the rate of convergence for the decomposition with respect to this basis. In the second main result we show that with additional assumptions the set of root vectors of the operator \(A\) is a Riesz basis for \(H\).
Keywords: nonhomogeneous damped string, Hilbert space, Riesz basis, modulus of continuity, basis with parentheses, basis of subspaces, string equation.
Mathematics Subject Classification: 34L10, 34B08.
- R.A. DeVore, G.G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin, 1993.
- J.B. Garnett, Bounded Analytic Functions - Revised First Edition, Graduate Texts in Mathematics, Springer, New York, 2007.
- I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Non-selfadjoint Operators in Hilbert Space, vol. 18, Transl. Math. Monographs, AMS, 1969.
- A.M. Gomilko, V.N. Pivovarchik, Asymptotics of solutions of the Sturm-Loiuville equation with respect to a parameter, Ukr. Matem. Zh. 53 (2001), 742-757 [in Russian]; English transl. in Ukrainian Math. J. 53 (2001), 866-885.
- A.M. Gomilko, Ł. Rzepnicki, Basis properties of eigenfunctions of the boundary problem of small vibrations of a nonhomogeneous string with damping at the end, Asymptotic Analysis 92 (2015), 107-140.
- Ł. Rzepnicki, Estimations of solutions of the Sturm-Liouville equation with respect to a spectral parameter, Integral Equations and Operator Theory 76 (2013), 565-588.
- Ł. Rzepnicki, Generating the exponentially stable \(C_0\)-semigroup in a nonhomogeneous string equation with damping at the end, Opuscula Math. 33 (2013) 1, 151-162.
- R. Shakarchi, E.M. Stein, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton Lectures in Analysis, Princeton University Press, Princeton, 2005.
- M.A. Shubov, Asymptotic and spectral analysis of non-selfadjoint operators generated by a filament model with a critical value of a boundary parameter, Math. Meth. Appl. Sci. 23 (2003), 213-245.
- M.A. Shubov, Nonselfadjoint operators generated by the equation of a nonhomogeneous damped string, Trans. Amer. Math. Soc. 349 (1997) 11, 4481-4499.
- M.A. Shubov, Nonhomogeneous Damped String: Riesz Basis Property of Root Vectors via Transformation Operators Method, Progress in Nonlinear Differential Equations and Their Applications, vol. 42, Birkhäuser Verlag, Basel, 2000.
- P.P. Zabrejko, A.I. Koshelev, M.A. Krasnosel'skii, S.G. Mikhlin, L.S. Rakovschik, V.J. Stetsenko, Integral Equations, Nordhoff, Leyden-Massachusetts, USA, 1975.