Equiconvergence theorems for Sturm--Lioville operators with singular potentials (rate of equiconvergence in $W_2^\theta$-norm)

I. V. Sadovnichaya

Eurasian mathematical journal, Tome 1 (2010) no. 1, pp. 137-146

Résumé

We study the Sturm–Liouville operator $Ly=l(y)=-\dfrac{d^2y}{dx^2}+q(x)y$ with Dirichlet boundary conditions $y(0)=y(\pi)=0$ in the space $L_2[0,\pi]$. We assume that the potential has the form $q(x)=u'(x)$, where $u\in W_2^{\theta}[0,\pi]$ with $0\theta1/2$. Here $W_2^{\theta}[0,\pi]=[L_2,W_2^1]_\theta$ is the Sobolev space. We consider the problem of equiconvergence in $W_2^\theta[0,\pi]$-norm of two expansions of a function $f\in L_2[0,\pi]$. The first one is constructed using the system of the eigenfunctions and associated functions of the operator $L$. The second one is the Fourier expansion in the series of sines. We show that the equiconvergence holds for any function $f$ in the space $L_2[0,\pi]$.