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

TL;DR

This paper establishes equiconvergence theorems for Sturm--Liouville operators with complex distribution potentials, demonstrating uniform convergence of eigenfunction series and providing estimates for the rate of convergence.

Contribution

It introduces new equiconvergence results for operators with distribution potentials in Sobolev spaces, including convergence rates and spectral properties.

Findings

Eigenfunctions form a Riesz basis in L2[0,π]

Series for eigenfunctions and sine functions are uniformly equiconvergent

Provides estimates for 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.