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Mathematical Notes

, Volume 104, Issue 5–6, pp 781–788 | Cite as

Optimal Recovery Methods for Solutions of the Dirichlet Problem that are Exact on Subspaces of Spherical Harmonics

  • E. A. BalovaEmail author
  • K. Yu. Osipenko
Article
  • 16 Downloads

Abstract

We consider the optimal recovery problem for the solution of the Dirichlet problem for the Laplace equation in the unit d-dimensional ball on a sphere of radius ρ from a finite collection of inaccurately specified Fourier coefficients of the solution on a sphere of radius r, 0 < r < ρ < 1. The methods are required to be exact on certain subspaces of spherical harmonics.

Keywords

optimal recovery Dirichlet problem Laplace equation spherical harmonics 

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References

  1. 1.
    S. M. Nikol’skii, “Concerning estimation for approximate quadrature formulas,” Uspekhi Mat. Nauk 5 (2 (36)), 165–177 (1950).MathSciNetGoogle Scholar
  2. 2.
    A. Sard, “Best approximative integration formulas; best approximation formulas,” Amer. J. Math. 71 (1), 80–91 (1949).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J. F. Traub and H. Woźniakowski, A General Theory of Optimal Algorithms (Academic Press, New York, 1980).zbMATHGoogle Scholar
  4. 4.
    L. Plaskota, Noisy Information and Computational Complexity (Cambridge Univ. Press, Cambridge, 1996).CrossRefzbMATHGoogle Scholar
  5. 5.
    K. Yu. Osipenko, Optimal Recovery of Analytic Functions (Nova Science Publ., Huntington, NY, 2000).Google Scholar
  6. 6.
    G. G. Magaril-Il’yaev and V. M. Tikhomirov, Convex Analysis and Its Applications (Éditorial URSS, Moscow, 2011) [in Russian].zbMATHGoogle Scholar
  7. 7.
    G. G. Magaril-Il’yaev and K. Yu. Osipenko, “Exactness and optimality of methods for recovering functions from their spectrum,” in Trudy Mat. Inst. Steklova, Vol. 293: Function Spaces, Approximation Theory, and Related Problems of Mathematical Analysis (MAIK Nauka/Interperiodica, Moscow, 2016), pp. 201–216 [Proc. Steklov Inst. Math. 293, 194–208 (2016)].Google Scholar
  8. 8.
    K. Yu. Osipenko, “On the reconstruction of the solution of the Dirichlet problem from inexact initial data,” Vladikavkaz.Mat. Zh. 6 (4), 55–62 (2004).MathSciNetzbMATHGoogle Scholar
  9. 9.
    E. A. Balova, “Optimal Reconstruction of the solution of the Dirichlet problem from inaccurate input data,” Mat. Zametki 82 (3), 323–334 (2007) [Math. Notes 82 (3), 285–294 (2007)].MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton Univ. Press, Princeton, NJ, 1971; Mir, Moscow, 1974).zbMATHGoogle Scholar
  11. 11.
    K. Yu. Osipenko, “Optimal recovery of linear operators in non-Euclidean metrics,” Mat. Sb. 205 (10), 77–106 (2014) [Sb.Math. 205 (10), 1442–1472 (2014)].MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Moscow Aviation Institute (National Research University)MoscowRussia
  2. 2.LomonosovMoscow State UniversityMoscowRussia
  3. 3.Kharkevich Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia

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