Unconditional basicity of eigenfunctions’ system of Sturm-Liouville operator with an involutional perturbation

Abstract

In this paper the question on unconditional basicity of the system of eigenfunctions of the involutive perturbed Sturm-Liouville operator is investigated. The Green's function of the operator under consideration in the case of constant coefficients is constructed. The estimates of the Green's functions are obtained. The existence of the Green's function is shown in the case when the operator under consideration has a variable coefficient. The theorem on the equiconvergence of expansions with respect to the eigenfunctions of the indicated operators is proved with the help of the Green's function. The basicity of the eigenfunctions of the operator under consideration in the class L2 (−1, 1) is proved. It is established that the basis from the eigenfunctions of the involutive perturbed Sturm-Liouville operator is the unconditional basis.

The Expansion Theorems for Sturm-Liouville Operators with an Involution Perturbation

“…For j = 0 the estimate ( 22) directly follows from Lemma 2 and the relation (21). Let us 148 assume that the assertion is true for j = n and prove its validity for j = n + 1.…”

“…Let us 148 assume that the assertion is true for j = n and prove its validity for j = n + 1. Using the 149 relations (20), (21) and Lemma 2 we obtain 150…”