Equiconvergence of spectral decompositions for the Dirac system with potential in Lebesgue spaces

Abstract

The problem of equiconvergence of spectral decompositions corresponding to the systems of root functions of two one-dimensional Dirac operators ℒ_P,U and ℒ_0,U with potential P summable on a finite interval and Birkhoff-regular boundary conditions is analyzed. It is proved that in the case of P ∈ L_ϰ[0, π], ϰ ∈ (1,∞], equiconvergence holds for every function f ∈ L_μ[0, π], μ ∈ [1,∞], in the norm of the space L_ν[0, π], ν ∈ [1,∞], if the indices ϰ, μ, and ν satisfy the inequality 1/ϰ + 1/μ − 1/ν ≤ 1 (except for the case when ϰ = ν = ∞ and μ = 1). In particular, in the case of a square summable potential, the uniform equiconvergence on the interval [0, π] is proved for an arbitrary function f ∈ L₂[0, π].