Opuscula Math. 37, no. 1 (2017), 141-165
http://dx.doi.org/10.7494/OpMath.2017.37.1.141

Opuscula Mathematica

# The basis property of eigenfunctions in the problem of a nonhomogeneous damped string

Łukasz Rzepnicki

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.

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• Łukasz Rzepnicki
• Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, ul. Chopina 12/18, 87-100 Toruń, Poland
• Communicated by P.A. Cojuhari.
• Revised: 2016-10-03.
• Accepted: 2016-10-03.
• Published online: 2016-12-14.