Journal of Mathematical Sciences

, Volume 155, Issue 1, pp 57–80 | Cite as

On a norm and approximate characteristics of classes of multivariable functions

Article

Abstract

We introduce a space of quasicontinuous functions and study its approximate characteristics, i.e., ε-entropy and widths. We establish inequalities for norms of trigonometric polynomials in this space. In addition, we obtain exponents of the ε-entropy and widths of some classes of functions with low smoothness.

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References

  1. 1.
    E. S. Belinskii, “Asymptotic characteristics for classes of functions with condition on mixed derivative (mixed difference),” in: Investigations on the Theory of Real Multivalued Functions, Yaroslavl (1990), p. 22–37.Google Scholar
  2. 2.
    E. S. Belinskiy, “Estimates of entropy numbers and Gaussian measures for classes of functions with bounded mixed derivative,” J. Approx. Theory, 93, No. 1, 114–127 (1998).CrossRefMathSciNetGoogle Scholar
  3. 3.
    S. V. Bochkarev, “De la Vallée-Poussin series in the BMO, L 1, H 1(D) spaces and multiplicative inequalities,” Tr. Mat. Inst. Steklova, 210, 41–64 (1995).MathSciNetGoogle Scholar
  4. 4.
    E. D. Gluskin, “Extreme properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces,” Mat. Sb., 136(178), No. 1, 85–96 (1988).Google Scholar
  5. 5.
    P. G. Grigoriev, “On a sequence of trigonometric polynomials,” Mat. Zametki, 61, No. 6, 935–938 (1997).Google Scholar
  6. 6.
    B. S. Kashin, “On some properties of a space of trigonometric polynomials with uniform norm,” Tr. Mat. Inst. Steklova, 145, 111–116 (1980).MATHMathSciNetGoogle Scholar
  7. 7.
    B. S. Kashin, “On properties of N-dimensional cube random sections,” Vestn. MGU, Ser. Mat., Mech., No. 3, 8–11 (1983).Google Scholar
  8. 8.
    B. S. Kashin and A. A. Saakian, Orthogonal Series [in Russian], Nauka, Moscow (1984).MATHGoogle Scholar
  9. 9.
    B. S. Kashin and V. N. Temlyakov, “On estimates for approximate characteristics of classes of functions with bounded mixed derivative,” Mat. Zametki, 58, No. 6, 922–925 (1995).MathSciNetGoogle Scholar
  10. 10.
    B. S. Kashin and V. N. Temlyakov, “On a norm and related applications,” Mat. Zametki, 64, No. 4, 637–640 (1998).MathSciNetGoogle Scholar
  11. 11.
    G. G. Lorentz, “Metric entropy and approximation,” Bull. Amer. Math. Soc., 72, 903–937 (1966).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    J. Marcinkiewicz, Collected Papers, PWN, Warszawa (1964).MATHGoogle Scholar
  13. 13.
    A. Pajor and N. Tomczak-Jaegermann, “Subspaces of small codimension of finite-dimensional Banach spaces,” Proc. Amer. Math. Soc., 97, No. 4, 637–642 (1986).MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    A. Pietsch, Operator Ideals [Russian translation], Mir, Moscow (1982).MATHGoogle Scholar
  15. 15.
    G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press, Cambridge (1989).MATHGoogle Scholar
  16. 16.
    C. Schütt, “Entropy numbers of diagonal operators between symmetric Banach spaces,” J. Approx. Theory, 40, 121–128 (1984).MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Z. Ŝidak, “Rectangular confidence regions for the means of multivariate normal distributions,” J. Amer. Statist. Assoc., 62, 626–633 (1967).MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    M. Talagrand, “The small ball problem for the Brownian sheet,” Ann. Probab., 22, 1331–1354 (1994).MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    V. N. Temlyakov, “An approximation of functions with bounded mixed derivative,” Tr. Mat. Inst. Steklova, 178 (1986).Google Scholar
  20. 20.
    V. N. Temlyakov, “Estimates for asymptotic characteristics of classes of functions with bounded mixed derivative or difference,” Tr. Mat. Inst. Steklova, 189, 138–168 (1989).MathSciNetGoogle Scholar
  21. 21.
    V. N. Temlyakov, “Estimates for the best bilinear approximations of functions and approximate numbers of integral operators,” Mat. Zametki, 51, No. 5, 125–134 (1992).MathSciNetGoogle Scholar
  22. 22.
    V. N. Temlyakov, “An inequality for trigonometric polynomials and its application for estimating the entropy numbers,” J. Complexity, 11, 293–307 (1995).MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    V. N. Temlyakov, “An inequality for trigonometric polynomials and its application for estimating the Kolmogorov widths,” East J. Approx., 2, No. 2, 253–262 (1996).MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Steklov Institute of MathematicsMoscowRussia
  2. 2.The University of South CarolinaColumbiaUSA

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