Doklady Mathematics

, Volume 86, Issue 1, pp 542–544 | Cite as

Equiconvergence of spectral decompositions of Hill operators

  • P. B. Djakov
  • B. S. Mityagin
Mathematics

Abstract

We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator L = −d2/dx2 + v(x), xL1([0, π], with Hper−1-potential and the free operator L0 = −d2/dx2, subject to periodic, antiperiodic or Dirichlet boundary conditions. In particular, we prove that
$$\left\| {S_N - S_N^0 :L^a \to L^b } \right\| \to 0if1 < a \leqslant b < \infty ,1/a - 1/b < 1/2,$$
, where SN and SN0 are the N-th partial sums of the spectral decompositions of L and L0. Moreover, if vH−α with 1/2 < α < 1 and \(\frac{1} {a} = \frac{3} {2} - \alpha \), then we obtain the uniform equiconvergence ‖SNSN0: LaL‖ → 0 as N → ∞.

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Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  • P. B. Djakov
    • 1
  • B. S. Mityagin
    • 1
  1. 1.Sabanci University, Istanbul, Turkey The Ohio State UniversityColumbusUSA

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