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

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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 ℒ0,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 ∈ L 2[0, π].

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  1. Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119991, Russia

    I. V. Sadovnichaya

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Original Russian Text © I.V. Sadovnichaya, 2016, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 293, pp. 296–324.

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Sadovnichaya, I.V. Equiconvergence of spectral decompositions for the Dirac system with potential in Lebesgue spaces. Proc. Steklov Inst. Math. 293, 288–316 (2016). https://doi.org/10.1134/S0081543816040209

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