The basis property in Lp of the boundary value problem rationally dependent on the eigenparameter

Abstract

We consider a Sturm–Liouville operator with boundary conditions rationally dependent on the eigenparameter. We study the basis property in Lp of the system of eigenfunctions corresponding to this operator. We determine the explicit form of the biorthogonal system. Using this we establish a theorem on the minimality of the part of the system of eigenfunctions. For the basisness in L2 we prove that the system of eigenfunctions is quadratically close to trigonometric systems. For the basisness in Lp we use F. Riesz’s theorem.