Equiconvergence of spectral decompositions of Hill operators

• Published in: Doklady. Mathematics 2012-07, Vol.86 (1), p.542-544
• Main Authors: Djakov, P. B., Mityagin, B. S.
• Format: Article
• Language: English
• Subjects: Mathematics; Mathematics and Statistics
• ISSN: 1064-5624; 1531-8362
• DOI: 10.1134/S1064562412040333

We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator L = − d 2 / dx 2 + v ( x ), x ∈ L 1 ([0, π], with H per −1 -potential and the free operator L 0 = − d 2 / dx 2 , subject to periodic, antiperiodic or Dirichlet boundary conditions. In particular, we prove that , where S N and S N 0 are the N -th partial sums of the spectral decompositions of L and L 0 . Moreover, if v ∈ H −α with 1/2 < α < 1 and , then we obtain the uniform equiconvergence ‖ S N − S N 0 : L a → L ∞ ‖ → 0 as N → ∞.