On the stability of basisness in Lp (1 p +∞) of cosines and sines

We study the basis properties in L_p(0, pi) (1 p infty) of the solution system of Sturm--Liouville equations with different types of initial conditions. We first establish some results on the stability of the basis property of cosines and sines in L_p(0, pi) (1 p infty) and then show that the solution system above forms a basis in L_p(0, pi) if and only if certain cosine system (or sine system, depending on type of initial conditions) forms a basis in L_p(0, pi).