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Wavelet approximation and Fourier widths of classes of periodic functions of several variables. I

  • D. B. BazarkhanovEmail author
Article

Abstract

We obtain characterizations (and prove the corresponding equivalence of norms) of function spaces B pq sm (\( \mathbb{I} \) k ) and L pq sm (\( \mathbb{I} \) k ) of Nikol’skii-Besov and Lizorkin-Triebel types, respectively, in terms of representations of functions in these spaces by Fourier series with respect to a multiple system \( \mathcal{W}_m^\mathbb{I} \) of Meyer wavelets and in terms of sequences of the Fourier coefficients with respect to this system. We establish order-sharp estimates for the approximation of functions in B pq sm (\( \mathbb{I} \)) and L pq sm (\( \mathbb{I} \) k ) by special partial sums of these series in the metric of L r (\( \mathbb{I} \) k ) for a number of relations between the parameters s, p, q, r, and m (s = (s 1, ..., s n ) ∈ ℝ + n , 1 ≤ p, q, r ≤ ∞, m = (m 1, ..., m n ) ∈ ℕ n , k = m 1 +... + m n , and \( \mathbb{I} \) = ℝ or \( \mathbb{T} \)). In the periodic case, we study the Fourier widths of these function classes.

Keywords

Function Space STEKLOV Institute Trigonometric Polynomial Steklov Inst Unconditional Basis 
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References

  1. 1.
    S. M. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems, 2nd ed. (Nauka, Moscow, 1977) [in Russian].Google Scholar
  2. 2.
    O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Imbedding Theorems, 2nd ed. (Nauka, Moscow, 1996) [in Russian].zbMATHGoogle Scholar
  3. 3.
    H. Triebel, Theory of Function Spaces (Birkhäuser, Basel, 1983; Mir, Moscow, 1986).CrossRefGoogle Scholar
  4. 4.
    H.-J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces (J. Wiley & Sons, Chichester, 1987).Google Scholar
  5. 5.
    T. I. Amanov, Spaces of Differentiable Functions with Dominating Mixed Derivative (Nauka, Alma-Ata, 1976) [in Russian].Google Scholar
  6. 6.
    V. N. Temlyakov, Approximation of Functions with a Bounded Mixed Derivative (Nauka, Moscow, 1986), Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 178 [Proc. Steklov Inst. Math. 178 (1989)].Google Scholar
  7. 7.
    V. N. Temlyakov, Approximation of Periodic Functions (Nova Sci. Publ., Commack, NY, 1993).zbMATHGoogle Scholar
  8. 8.
    H.-J. Schmeisser, “On Spaces of Functions and Distributions with Mixed Smoothness Properties of Besov-Triebel-Lizorkin Type. I, II,” Math. Nachr. 98, 233–250 (1980); 106, 187–200 (1982).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    M. Yamazaki, “Boundedness of Product Type Pseudodifferential Operators on Spaces of Besov Type,” Math. Nachr. 133, 297–315 (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    H.-J. Schmeisser, “Recent Developments in the Theory of Function Spaces with Dominating Mixed Smoothness,” in Nonlinear Analysis, Function Spaces and Applications: Proc. Spring School, Prague, May 30–June 6, 2006 (Czech Acad. Sci., Math. Inst., Prague, 2007), Vol. 8, pp. 145–204.Google Scholar
  11. 11.
    D. B. Bazarkhanov, “Characterizations of the Nikol’skii-Besov and Lizorkin-Triebel Function Spaces of Mixed Smoothness,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 243, 53–65 (2003) [Proc. Steklov Inst. Math. 243, 46–58 (2003)].MathSciNetGoogle Scholar
  12. 12.
    D. B. Bazarkhanov, “ϕ-Transform Characterization of the Nikol’skii-Besov and Lizorkin-Triebel Function Spaces with Mixed Smoothness,” East J. Approx. 10(1–2), 119–131 (2004).zbMATHMathSciNetGoogle Scholar
  13. 13.
    D. B. Bazarkhanov, “Equivalent (Quasi)norms for Certain Function Spaces of Generalized Mixed Smoothness,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 248, 26–39 (2005) [Proc. Steklov Inst. Math. 248, 21–34 (2005)].MathSciNetGoogle Scholar
  14. 14.
    D. B. Bazarkhanov, “Different Representations and Equivalent Norms for Nikol’skii-Besov and Lizorkin-Triebel Spaces of Generalized Mixed Smoothness,” Dokl. Akad. Nauk 402(3), 298–302 (2005) [Dokl. Math. 71 (3), 373–377 (2005)].MathSciNetGoogle Scholar
  15. 15.
    J. Vybiral, Function Spaces with Dominating Mixed Smoothness (Inst. Math., Pol. Acad. Sci., Warszawa, 2006), Diss. Math. 436.Google Scholar
  16. 16.
    M. Hansen and J. Vybiral, “The Jawerth-Franke Embedding of Spaces with Dominating Mixed Smoothness,” Georgian Math. J. 16(4), 667–682 (2009).zbMATHMathSciNetGoogle Scholar
  17. 17.
    Y. Meyer, Wavelets and Operators (Cambridge Univ. Press, Cambridge, 1992), Cambridge Stud. Adv. Math. 37.zbMATHGoogle Scholar
  18. 18.
    B. S. Kashin and A. A. Saakyan, Orthogonal Series, 2nd ed. (AFTs, Moscow, 1999) [in Russian].zbMATHGoogle Scholar
  19. 19.
    P. Wojtaszczyk, A Mathematical Introduction to Wavelets (Cambridge Univ. Press, Cambridge, 1997).zbMATHCrossRefGoogle Scholar
  20. 20.
    E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton Univ. Press, Princeton, NJ, 1971; Mir, Moscow, 1974).zbMATHGoogle Scholar
  21. 21.
    E. Hernández and G. Weiss, A First Course on Wavelets (CRC Press, Boca Raton, FL, 1996), Stud. Adv. Math.zbMATHCrossRefGoogle Scholar
  22. 22.
    H. Triebel, Theory of Function Spaces. III (Birkhäuser, Basel, 2006).zbMATHGoogle Scholar
  23. 23.
    C. Fefferman and E. M. Stein, “Some Maximal Inequalities,” Am. J. Math. 93, 107–115 (1971).zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (Princeton Univ. Press, Princeton, NJ, 1993).zbMATHGoogle Scholar
  25. 25.
    R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov, “Hyperbolic Wavelet Approximation,” Constr. Approx. 14(1), 1–26 (1998).CrossRefMathSciNetGoogle Scholar
  26. 26.
    H. Wang, “Representation and Approximation of Multivariate Functions with Mixed Smoothness by Hyperbolic Wavelets,” J. Math. Anal. Appl. 291, 698–715 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    P. I. Lizorkin, “On Bases and Multipliers in the Spaces B rp,θ(Γ),” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 143, 88–104 (1977) [Proc. Steklov Inst. Math. 143, 93–110 (1980)].MathSciNetGoogle Scholar
  28. 28.
    D. G. Orlovskiĭ, “On Multipliers in the Space B p,θr,” Anal. Math. 5, 207–218 (1979).zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    H.-J. Schmeisser, “An Unconditional Basis in Periodic Spaces with Dominating Mixed Smoothness Properties,” Anal. Math. 13, 153–168 (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Dinh Dũng, “Stability in Periodic Multi-wavelet Decompositions and Recovery of Functions,” East J. Approx. 11(4), 447–479 (2005).zbMATHMathSciNetGoogle Scholar
  31. 31.
    A. V. Andrianov and V. N. Temlyakov, “On Two Methods of Generalization of Properties of Univariate Function Systems to Their Tensor Product,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 219, 32–43 (1997) [Proc. Steklov Inst. Math. 219, 25–35 (1997)].MathSciNetGoogle Scholar
  32. 32.
    É. M. Galeev, “Approximation of Classes of Periodic Functions of Several Variables by Nuclear Operators,” Mat. Zametki 47(3), 32–41 (1990) [Math. Notes 47, 248–254 (1990)].MathSciNetGoogle Scholar
  33. 33.
    É. M. Galeev, “Widths of the Besov Classes B p,θr(\( \mathbb{T} \) d),” Mat. Zametki 69(5), 656–665 (2001) [Math. Notes 69, 605–613 (2001)].MathSciNetGoogle Scholar
  34. 34.
    A. S. Romanyuk, “Best Approximations and Widths of Classes of Periodic Functions of Several Variables,” Mat. Sb. 199(2), 93–114 (2008) [Sb. Math. 199, 253–275 (2008)].MathSciNetGoogle Scholar
  35. 35.
    H.-J. Schmeisser and W. Sickel, “Spaces of Functions of Mixed Smoothness and Approximation from Hyperbolic Crosses,” J. Approx. Theory 128, 115–150 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    T. Ullrich, “Smolyak’s Algorithm, Sampling on Sparse Grids and Sobolev Spaces of Functions of Dominating Mixed Smoothness,” East J. Approx. 14, 1–38 (2008).MathSciNetGoogle Scholar
  37. 37.
    D. B. Bazarkhanov, “Estimates of the Fourier Widths of Classes of Nikol’skii-Besov and Lizorkin-Triebel Types of Periodic Functions of Several Variables,” Mat. Zametki 87(2), 305–308 (2010) [Math. Notes 87, 281–284 (2010)].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Institute of MathematicsAlmatyKazakhstan

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