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Best trigonometric and bilinear approximations of classes of functions of several variables

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Abstract

Order-sharp estimates of the best orthogonal trigonometric approximations of the Nikol’skii-Besov classes B r p,θ of periodic functions of several variables in the space L q are obtained. Also the orders of the best approximations of functions of 2d variables of the form g(x, y) = f(x−y), x, y ∈ \(\mathbb{T}\) d = Π d j=1 [−π, π], f(x) ∈ B r p,θ , by linear combinations of products of functions of d variables are established.

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References

  1. O. V. Besov, “On some families of the function spaces. Imbedding and continuation theorems,” Dokl. Akad. Nauk SSSR 126(6), 1163–1165 (1959).

    MathSciNet  MATH  Google Scholar 

  2. S. M. Nikol’skii, “Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of several variables,” in Trudy Mat. Inst. Steklov Collection of papers dedicated to the 60th anniversary of Academician I. I. Vinogradov (Izd. AN SSSR, Moscow, 1951), Vol. 38, pp. 244–278 [in Russian].

    Google Scholar 

  3. P. I. Lizorkin, “Generalized Holder spaces B (r) p.Θ and their relation with the Sobolev spaces L (r)p ,” Sibirsk. Mat. Zh. 9(5), 1127–1152 (1968).

    MathSciNet  MATH  Google Scholar 

  4. A. S. Romanyuk, “Approximation classes of periodic functions of several variables,” Mat. Zametki 71(1), 109–121 (2002) [Math. Notes 71 (1), 98–109 (2002)].

    Article  MathSciNet  Google Scholar 

  5. A. S. Romanyuk, “Bilinear and trigonometric approximations of periodic functions of several variables of Besov classes B r p,θ ,” Izv. Ross. Akad. Nauk Ser. Mat. 70(2), 69–98 (2006) [Izv. Math. 70 (2), 277–306 (2006)].

    Article  MathSciNet  Google Scholar 

  6. S. B. Stechkin, “On absolute convergence of orthogonal series,” Dokl. Akad. Nauk SSSR 102(1), 37–40 (1955).

    MathSciNet  MATH  Google Scholar 

  7. A. S. Romanyuk, “Best M-term trigonometric approximations of Besov classes of periodic functions of several variables,” Izv. Ross. Akad. Nauk Ser. Mat. 67(2), 61–100 (2003) [Izv. Math. 67 (2), 265–302 (2003)].

    Article  MathSciNet  Google Scholar 

  8. R. A. DeVore and V. N. Temlyakov, “Nonlinear approximation by trigonometric sums,” J. Fourier Anal. Appl. 2(1), 29–48 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  9. N. P. Korneichuk, Sharp Constants in Approximation Theory (Nauka, Moscow, 1987) [in Russian].

    Google Scholar 

  10. V. N. Temlyakov, “Approximations of functions with bounded mixed derivative,” in Trudy Mat. Inst. Steklov (Nauka, Moscow, 1986), Vol. 178, pp. 3–113 [Proc. Steklov Inst. Math. 178, 1–121 (1989].

    Google Scholar 

  11. V. N. Temlyakov, “Greedy algorithm and m-term trigonometric approximation,” Constr. Approx. 14(4), 569–587 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  12. E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen,” Math. Ann. 63(4), 433–476 (1907).

    Article  MathSciNet  MATH  Google Scholar 

  13. V. N. Temlyakov, “Bilinear approximation and applications,” in Trudy Mat. Inst. Steklov Vol. 187: Studies in the Theory of Differentiable Functions of Several Variables and Its Applications. Part 13, Collection of papers (Nauka, Moscow, 1989), pp. 191–215 [Proc. Steklov Inst.Math. 187, 221–248 (1990)].

    Google Scholar 

  14. V. N. Temlyakov, “Bilinear approximation and related questions,” in Trudy Mat. Inst. Steklov Vol. 194: Studies in the Theory of Differentiable Functions of Several Variables and Its Applications. Part 14 (Nauka, Moscow, 1992), pp. 229–248 [Proc. Steklov Inst. Math. 187, 245–265 (1993)].

    Google Scholar 

  15. V. N. Temlyakov, “Approximation periodic functions of several variables by combinations of the functions depending on fewer variables,” in Trudy Mat. Inst. Steklov Vol. 173: Studies in the Theory of Differentiable Functions of Several Variables and Its Applications. Part 11, Collection of papers (Nauka, Moscow, 1986), pp. 243–252 [Proc. Steklov Inst. Math. 173, 263–272 (1987)].

    Google Scholar 

  16. B. S. Kashin and A. A. Saakyan, Orthogonal Series (Nauka, Moscow, 1984) [in Russian].

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine

    A. S. Romanyuk

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  1. A. S. Romanyuk
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Correspondence to A. S. Romanyuk.

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Original Russian Text © A. S. Romanyuk, 2013, published in Matematicheskie Zametki, 2013, Vol. 94, No. 3, pp. 401–415.

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Romanyuk, A.S. Best trigonometric and bilinear approximations of classes of functions of several variables. Math Notes 94, 379–391 (2013). https://doi.org/10.1134/S0001434613090095

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