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On the Riesz Basis Property of the Eigen- and Associated Functions of Periodic and Antiperiodic Sturm–Liouville Problems
O. A. Velieva,
A. A. Shkalikovb a Dogus University
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
The paper deals with the Sturm-Liouville operator
$$
Ly=-y''+q(x)y, \qquad x\in[0,1],
$$
generated in the space
$L_2=L_2[0,1]$ by periodic or antiperiodic boundary conditions. Several theorems on the Riesz basis property of the root functions of the operator
$L$ are proved. One of the main results is the following. Let
$q$ belong to the Sobolev space
$W_1^p[0,1]$ for some integer
$p\ge0$ and satisfy the conditions
$q^{(k)}(0)=q^{(k)}(1)=0$ for
$0\le k\le s-1$, where
$s\le p$. Let the functions
$Q$ and
$S$ be defined by the equalities
$$
Q(x)=\int_0^xq(t)\,dt,\qquad S(x)=Q^2(x)
$$
and let
$q_n$,
$Q_n$, and
$S_n$ be the Fourier coefficients of
$q$,
$Q$, and
$S$ with respect to the trigonometric system
$\{e^{2\pi inx}\}_{-\infty}^\infty$. Assume that the sequence
$q_{2n}-S_{2n}+2Q_0Q_{2n}$ decreases not faster than the powers
$n^{-s-2}$. Then the system of eigenfunctions and associated functions of the operator
$L$ generated by periodic boundary conditions forms a Riesz basis in the space
$L_2[0,1]$ (provided that the eigenfunctions are normalized) if and only if the condition
$$
q_{2n}-S_{2n}+2Q_0Q_{2n}\asymp q_{-2n}-S_{-2n}+2Q_0Q_{-2n},\qquad n>1,
$$
holds.
Keywords:
periodic Sturm-Liouville problem, Hill operator, Riesz basis, Sobolev spaces, Birkhoff regularity, Fourier series, Jordan chain.
UDC:
517.984 Received: 20.02.2008
Revised: 30.10.2008
DOI:
10.4213/mzm6912
Fulltext:
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