Equiconvergence theorems for Sturm--Liouville operators with distribution potentials^ the rate of equiconvergence

Abstract

We consider a Sturm--Liouville operator $Ly=-y''+qy$ in $L_2[0,\pi]$ with Dirichlet boundary conditions. We assume, that the potential $q$ is complex valued and belongs to Sobolev space $W_2^\theta[0,\pi]$, $\theta\in(-1,-1/2$. This operators were successfully defined in papers of Savchuk A.M. and Shkalikov A.A. There were also shown, that theese operators have a discrete spectrum, which we denote by $\{\lambda_n\}$, and $\lim\lambda_n=+\infty$. All but finitely many of them are simple. The eigenfunctions form the Riesz basis in $L_2[0,\pi]$. We investigate a uniform on $[0,\pi]$ equiconvergence of series for this system and for trigonometric system $\{\sin(nt)\}_1^\infty$. We obtain not only a theorems of equiconvergence, but also estimate a rate of this equiconvergence.