We present a description of the relationship between the Sobolev spaces \( {W}_2^1 \)(ℝ) and \( {W}_2^2 \)(ℝ) and the Hilbert space ℓ2. Let Y be a finite or countable set of points on ℝ and let d ≔ inf {|y′ − y′′|, y′, y′′ ∈ Y, y′ ≠ y′′}. By using this relationship, we prove that if d = 0, then the systems of δ –functions {δ(x − yj), yj ∈ Y} and their derivatives {δ′(x − yj), yj ∈ Y} do not form Riesz bases in the closures of their linear spans in the Sobolev spaces \( {W}_2^{-1} \)(ℝ) and \( {W}_2^{-2} \)(ℝ) but, on the contrary, form the indicated bases in the case where d > 0. We also present the description of Friedrichs and Krein extensions and demonstrate their transversality. Moreover, the construction of a basis boundary triple and the description of all nonnegative self-adjoint extensions of the operator A′ are presented.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 12, pp. 1615–1624, December, 2017.
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Koval’ov, Y.H. Point Interactions in a Line and the Riesz Bases of δ-Functions. Ukr Math J 69, 1877–1890 (2018). https://doi.org/10.1007/s11253-018-1477-0
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DOI: https://doi.org/10.1007/s11253-018-1477-0