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Convergence of cubature formulas of high trigonometric precision in multidimensional periodic sobolev spaces


We establish convergence in a norm of cubature formulas of high trigonometric precision on multidimensional periodic Sobolev spaces including spaces of fractional smoothness. The main result is obtained under the conventional conditions on smoothness of the space of integrands and distribution of the nodes of the cubature formulas.

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  1. 1.

    H. Bateman and A. Erdélyi, Higher Transcendental Functions. Elliptic and Modular Functions. Laméand Mathieu Functions (Nauka, Moscow, 1967) [in Russian].

    Google Scholar 

  2. 2.

    M. V. Noskov and H. J. Schmid, “Cubature formulas of high trigonometric accuracy,” Zh. Vychisl. Mat. Mat. Fiz. 44, 786 (2004) [Comput. Math. Math. Phys. 44, 740 (2004)].

    MATH  MathSciNet  Google Scholar 

  3. 3.

    N. N. Osipov, “Construction of lattice rules with a trigonometric d-property on the basis of extreme lattices,” Zh. Vychisl. Mat. Mat. Fiz. 48, 212 (2008) [Comput. Math. Math. Phys. 48, 201 (2008)].

    MATH  Google Scholar 

  4. 4.

    V. I. Polovinkin, “Sequences of functionals with a boundary layer,” Sib. Mat. Zh. 15, 413 (1974) [Sib. Math. J. 15, 296 (1974)].

    MATH  MathSciNet  Article  Google Scholar 

  5. 5.

    S. L. Sobolev, Introduction to the Theory of Cubature Formulas (Nauka, Moscow, 1974) [Cubature Formulas and Modern Analysis. An Introduction (Gordon and Breach Science Publishers, Montreux, 1992)].

    Google Scholar 

  6. 6.

    S. L. Sobolev and V. L. Vaskevich, The Theory of Cubature Formulas (Sobolev Institute of Mathematics, Novosibirsk, 1996; Kluwer Academic Publishers Group, Dordrecht, 1997).

    Google Scholar 

  7. 7.

    H. Triebel, “Sampling numbers and embedding constants,” Tr. Mat. Inst. Steklova. 248, 275 (2005) [Proc. Steklov Inst. Math., Mo. 1 (248), 268 (2005). ]

    MathSciNet  Google Scholar 

  8. 8.

    V. L. Vaskevich, “Embedding constants for periodic Sobolev spaces of fractional order,” Sib. Mat. Zh. 49, 1019 (2008) [Sib. Math. J. 49, 806 (2008)].

    MATH  MathSciNet  Article  Google Scholar 

  9. 9.

    V. L. Vaskevich, “Embedding constants and embedding functions for Sobolev-like spaces on the unit sphere,” Dokl. Akad. Nauk 433, 441 (2010) [Dokl. Math. 82, 568 (2010)].

    MathSciNet  Google Scholar 

  10. 10.

    V. L. Vaskevich, “The error and guaranteed accuracy of cubature formulas in multidimensional periodic Sobolev spaces,” Sib. Mat. Zh. 55, 971 (2014) [Sib. Math. J. 55, 792 (2014)].

    MathSciNet  Article  Google Scholar 

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Correspondence to V. L. Vaskevich.

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Original Russian Text © V.L. Vaskevich, 2015, published in Matematicheskie Trudy, 2015, Vol. 18, No. 1, pp. 3–14.

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Vaskevich, V.L. Convergence of cubature formulas of high trigonometric precision in multidimensional periodic sobolev spaces. Sib. Adv. Math. 25, 297–304 (2015).

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  • cubature formula
  • formula of high trigonometric precision
  • error function
  • periodic Sobolev space
  • embedding functions and constants