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

2008/03/21 by I. V. Sadovnichaya, Sadovnichaya, I. V.
Mathematics · #34L10 #47E05 #Algebraic and Geometric Analysis #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.0803.3166

openalex publication_date 2008/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a Sturm--Liouville operator Ly=-y''+qy in L₂[0,π] with Dirichlet boundary conditions. We assume, that the potential q is complex valued and belongs to Sobolev space W₂^θ[0,π], θ∈(-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 \λ_n\, and limλ_n=+∞. All but finitely many of them are simple. The eigenfunctions form the Riesz basis in L₂[0,π]. We investigate a uniform on [0,π] equiconvergence of series for this system and for trigonometric system \sin(nt)\₁^∞. We obtain not only a theorems of equiconvergence, but also estimate a rate of this equiconvergence.

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