Uniconvergence theorems for Sturm--Liouville operators with potentials from Sobolev space $W_2^{-1}[0,\pi ]$

Abstract

We consider a Sturm--Liouville $Ly=-y^{\prime \prime }+q(x)y$ in space $L_2[0,\pi ]$ with potential from Sobolev space $W_2^{-1}[0,\pi ]$. Moreover, we assume, that $q=u^{\prime }$, where $u\in L_2[0,\pi ]$. We consider Direchlet boundary conditions $y(0)=y(\pi )=0$, although we can treat a boundary conditions of Sturm type. It is known, that operators of such class have a discrete spectr with only accumulation point $+\infty$ and the system $\{y_k{\}}_1^{\infty }$ of eigen and associated functions is a Riesz basis in $L_2[0,\pi ]$. Moreover, this basis is a Hilbert--Schmidt perturbation of the basis $\{sin(kx){\}}_1^{\infty }$. In this paper we prove the uniconvergence theorem: for any element $f\in L_2[0,\pi ]$ the sequence $P_{n}f-S_{n}f\to 0$ as $n\to \infty$ in $C[0,\pi ]$ (here $P_n$ and $S_n$ are the Riesz projectors to $\{y_k{\}}_1^n$ and $\{\sin (kt){\}}_1^n$ respectively).

15 pages