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Peer-reviewedCC BYResults in Applied Mathematics

Optimal interpolation formulas in the Sobolev space of periodic functions

  • Kholmat Shadimetov1
  • Khurshidjon Khayatov2
  1. 1Tashkent State Transport University, 1, Odilkhodjaev str., Tashkent, 100167, Uzbekistan; V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 9, University str., Tashkent, 100174, Uzbekistan
  2. 2Bukhara State University, 11, M.Ikbol str., Bukhara, 200114, Uzbekistan; Corresponding author.
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Abstract

The problem of finding the extremal function of an interpolation formula and of evaluating the norm of the error functional in the Sobolev space was first posed by S.L.Sobolev. In that work (Sobolev, 1961 [1] ), the extremal function of the interpolation formula was obtained explicitly in the Sobolev space whose generalized derivatives of order m are square-integrable. The present paper considers a new optimal interpolation formula in the Sobolev space: using the discrete analog of the differential operator, we obtain the optimal coefficients of the interpolation formula for an arbitrary m , and the norm of the error functional is computed. Integration of the interpolant leads to a quadrature formula; three numerical examples confirm the theoretical order of convergence.

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Cite

BibTeXRIS
BibTeX
@article{shadimetov2026optimal,
  title = {{Optimal interpolation formulas in the Sobolev space of periodic functions}},
  author = {Kholmat Shadimetov and Khurshidjon Khayatov},
  journal = {Results in Applied Mathematics},
  year = {2026},
  month = oct,
  doi = {10.1016/j.rinam.2026.100786},
  url = {https://doi.org/10.1016/j.rinam.2026.100786},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Optimal interpolation formulas in the Sobolev space of periodic functions
AU  - Kholmat Shadimetov
AU  - Khurshidjon Khayatov
PY  - 2026
DA  - 2026/10/30
JO  - Results in Applied Mathematics
DO  - 10.1016/j.rinam.2026.100786
UR  - https://doi.org/10.1016/j.rinam.2026.100786
AB  - The problem of finding the extremal function of an interpolation formula and of evaluating the norm of the error functional in the Sobolev space was first posed by S.L.Sobolev. In that work (Sobolev, 1961 [1] ), the extremal function of the interpolation formula was obtained explicitly in the Sobolev space whose generalized derivatives of order m are square-integrable. The present paper considers a new optimal interpolation formula in the Sobolev space: using the discrete analog of the differential operator, we obtain the optimal coefficients of the interpolation formula for an arbitrary m , and the norm of the error functional is computed. Integration of the interpolant leads to a quadrature formula; three numerical examples confirm the theoretical order of convergence.
SN  - 2590-0374
ER  -