TY  - JOUR
ID  - Rzepnicki2017
LB  - Rzepnicki2017
AU  - Rzepnicki, Łukasz
TI  - The basis property of eigenfunctions in the problem of a nonhomogeneous damped string
ST  - The basis property of eigenfunctions in the problem of a nonhomogeneous damped string
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2017
VL  - 37
IS  - 1
SP  - 141
EP  - 165
DO  - http://dx.doi.org/10.7494/OpMath.2017.37.1.141
UR  - http://dx.doi.org/10.7494/OpMath.2017.37.1.141
AB  - 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 \(H\).
KW  - nonhomogeneous damped string
KW  - Hilbert space
KW  - Riesz basis
KW  - modulus of continuity
KW  - basis with parentheses
KW  - basis of subspaces
KW  - string equation
SN  - 1232-9274
ER  -