Affiliations: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland. E-mails: [email protected], [email protected] | Institute of Telecommunications and Global Information Space, National Academy of Sciences of Ukraine, Kyiv, Ukraine
Note: [] Corresponding author: Łukasz Rzepnicki, Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland. E-mail: [email protected]
Abstract: This paper is concerned with the equation of a nonhomogeneous string of length one with one end fixed and the other one damped with a parameter h∈C. This problem can be rewritten as an abstract Cauchy problem for a densely defined closed operator iAh acting on an appropriate energy Hilbert space H. Under assumptions that the density function of the string ρ∈W21[0,1] is strictly positive and has ρ(1)≠h2 (if h∈R), we prove that the set of root vectors of Ah form a basis with parentheses in H. We show that with the additional condition ∫01ω12(ρ′,τ)/τ2 dτ<∞, where ω1 is the integral modulus of continuity, the root vectors of the operator Ah form a Riesz basis in H.
Keywords: nonhomogeneous damped string, Hilbert space, Riesz basis, modulus of continuity, basis with parentheses
