%0 Journal Article
%T On Summability of Spectral Expansions Corresponding to the Sturm-Liouville Operator
%A Alexander S. Makin
%J International Journal of Mathematics and Mathematical Sciences
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/843562
%X We study the completeness property and the basis property of the root function system of the Sturm-Liouville operator defined on the segment [0, 1]. All possible types of two-point boundary conditions are considered. 1. Introduction The spectral theory of two-point differential operators was begun by Birkhoff in his two papers [1, 2] of 1908 where he introduced regular boundary conditions for the first time. It was continued by Tamarkin [3, 4] and Stone [5, 6]. Afterwards their investigations were developed in many directions. There is an enormous literature related to the spectral theory outlined above, and we refer to [7¨C18] and their extensive reference lists for this activity. The present communication is a brief survey of results in the spectral theory of the Sturm-Liouville operator: with two-point boundary conditions where the are linearly independent forms with arbitrary complex-valued coefficients and is an arbitrary complex-valued function of class . Our main focus is on the non-self-adjoint case. We will study the completeness property and the basis property of the root function system of operator (1.1), (1.2). The convergence of spectral expansions is investigated only in classical sense; that is, the question about the summability of divergent series by a generalized method is not considered. 2. Preliminaries Let us present briefly the main definitions and facts which will be used in what follows. Let be a Banach space with the norm , and let be its dual with the norm . A system of elements is said to be closed in if the linear span of this system is everywhere dense in ; that is, any element of the space can be approximated by a linear combination of elements of this system with any accuracy in the norm of the space . A system of elements is said to be minimal in if none of its elements belongs to the closure of the linear span of the other elements of this system. Theorem 2.1 (see [19]). A system is minimal if and only if there exists a biorthogonal system dual to it, that is, a system of linear functionals from such that for all . Moreover, if the initial system is simultaneously closed and minimal in , then the system biorthogonally dual to it is uniquely defined. We say that a system is uniformly minimal in , if there exists such that for all , where is the closure of the linear span of all elements with serial numbers . Theorem 2.2 (see [19]). A closed and minimal system is uniformly minimal in if and only if: A system forms a basis of the space if, for any element , there exists a unique expansion of it in the elements of the system,
%U http://www.hindawi.com/journals/ijmms/2012/843562/