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Ukrainian Mathematical Journal

, Volume 65, Issue 12, pp 1862–1882 | Cite as

Best Bilinear Approximations for the Classes of Functions of Many Variables

  • A. S. Romanyuk
  • V. S. Romanyuk
Article
  • 35 Downloads

We obtain upper bounds for the values of the best bilinear approximations in the Lebesgue spaces of periodic functions of many variables from the Besov-type classes. In special cases, it is shown that these bounds are order exact.

Keywords

Periodic Function Trigonometric Polynomial Lebesgue Space Order Exact Bilinear Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    O. V. Besov, “On a family of functional spaces in connection with embedding and continuity theorems,” Tr. Mat. Inst. Akad. Nauk SSSR, 60, 42–81 (1961).zbMATHMathSciNetGoogle Scholar
  2. 2.
    S. M. Nikol’skii, “Inequalities for entire functions of finite degree and their applications in the theory differentiable functions of many variables,” Tr. Mat. Inst. Akad. Nauk SSSR, 38, 244–278 (1951).Google Scholar
  3. 3.
    T. I. Amanov, “Representation and embedding theorems for the functional spaces S p,θ(r) B( n) and S p,θ(r) ∗ B,Tr. Mat. Inst. Akad. Nauk SSSR, 77, 5–34 (1965).zbMATHMathSciNetGoogle Scholar
  4. 4.
    P. I. Lizorkin and S. M. Nikol’skii, “Spaces of functions of mixed smoothness from the decomposition point of view” Tr. Mat. Inst. Akad. Nauk SSSR, 187, 143–161 (1989).MathSciNetGoogle Scholar
  5. 5.
    A. S. Romanyuk and V. S. Romanyuk, “Best bilinear approximations of functions from Nikol’skii–Besov classes,” Ukr. Mat. Zh., 64, No. 5, 685–697 (2012); English translation: Ukr. Math. J., 64, No. 5, 781–796 (2012).CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    V. N. Temlyakov, “Approximation of periodic functions of many variables by combinations of functions depending on smaller numbers of variables,” Tr. Mat. Inst. Akad. Nauk SSSR, 173, 243–252 (1986).MathSciNetGoogle Scholar
  7. 7.
    V. N. Temlyakov, “Estimates for the best bilinear approximations of periodic functions,” Tr. Mat. Inst. Akad. Nauk SSSR, 181, 250–267 (1988).zbMATHMathSciNetGoogle Scholar
  8. 8.
    V. N. Temlyakov, Approximation of Periodic Functions, Nova Science, New York (1993).zbMATHGoogle Scholar
  9. 9.
    S. M. Nikol’skii, Approximation of Functions of Many Variables and Embedding Theorems [in Russian], Nauka, Moscow (1969).Google Scholar
  10. 10.
    A. S. Romanyuk, “Bilinear and trigonometric approximations of the Besov classes periodic functions of many variables,” Izv. Ros. Akad. Nauk, Ser. Mat., 70, No. 2, 69–98 (2006).CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • A. S. Romanyuk
    • 1
  • V. S. Romanyuk
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKievUkraine

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