Equiconvergence Property for Spectral Expansions Related to Perturbations of the Operator −u''(−x) with Initial Data

Abstract

Uniform equiconvergence of spectral expansions related to the second-order differential operators with involution: $-u''(-x)$ and $-u''(-x)+q(x)u(x)$ with the initial data $u(-1)=0$, $u'(-1)=0$ is obtained. Starting with the spectral analysis of the unperturbed operator, the estimates of the Green's functions are established and then applied in the contour integrating approach to the spectral expansions. As a corollary, it is proved that the root functions of the perturbed operator form the basis in $L_2(-1,1)$ for any complex-valued coefficient $q(x)\in L_2(-1,1)$.