Equiconvergence theorems for Sturm–Lioville operators with singular potentials (rate of equiconvergence in W^θ_2-norm)

Abstract

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<\theta<1/2. 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 case the equiconvergence holds for any

function f in the space L_2[0,\pi].