On the Riesz Basis Property of the Eigen- and Associated Functions of   Periodic and Antiperiodic Sturm-Liouville Problems
  The paper deals with the Sturm-Liouville operator $$ Ly=-y^{\prime\prime}+q(x)y,\qquad x\in\lbrack0,1], $$ generated in the space $L_{2}=L_{2}[0,1]$ by periodic or antiperiodic boundary conditions. Several theorems on Riesz basis property of the root functions of the operator $L$ are proved. One of the main results is the following. \textsl{Let $q$ belong to Sobolev space $W_{1}^{p}[0,1]$ with some integer $p\geq0$ and satisfy the conditions $q^{(k)}(0)=q^{(k)}(1)=0$ for $0\leq k\leq s-1$, where s}$\leq p.$ \textsl{Let the functions $Q$ and $S$ be defined by the equalities $Q(x)=\int_{0}^{x}q(t) dt, S(x)=Q^{2}(x)$ and let $q_{n}%, Q_{n},S_{n}$ be the Fourier coefficients of $q,Q,S$ with respect to the trigonometric system $\{e^{2\pi inx}\}_{-\infty}^{\infty}$. Assume that the sequence $q_{2n}-S_{2n}+2Q_{0}Q_{2n}$ decreases not faster than the powers $n^{-s-2}$. Then the system of eigen 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 $$ q_{2n}-S_{2n}+Q_{0}Q_{2n}\asymp q_{-2n}-S_{-2n}+2Q_{0}Q_{-2n},\quad n>1, $$ holds. 
A.A.Shkalikov and O.A.Veliev