Journal of Mathematical Sciences

, Volume 238, Issue 4, pp 471–483 | Cite as

Gaussian Approximation Numbers and Metric Entropy

  • T. KühnEmail author
  • W. Linde

The aim of this paper is to survey properties of Gaussian approximation numbers. We state the basic relations between these numbers and other s-numbers as, e.g., entropy, approximation, or Kolmogorov numbers. Furthermore, we fill a gap and prove new two-sided estimates in the case of operators with values in a K-convex Banach space. In the final section, we apply relations between Gaussian and other s-numbers to the d-dimensional integration operator defined on L2[0, 1]d.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Artstein, V. D. Milman, and S. Z. Szarek, “Duality of metric entropy,” Ann. Math., 159, 1313–1328 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    D. Bilyk, M. T. Lacey, and A. Vagharshakyan, “On the small ball inequality in all dimensions,” J. Funct. Anal., 254, 2470–2502 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    B. Carl, “Inequalities of Bernstein–Jackson-type and the degree of compactness of operators in Banach spaces,” Ann. Inst. Fourier, 35, 79–118 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    B. Carl, I. Kyrezi, and A. Pajor, “Metric entropy of convex hulls in Banach spaces,” J. London Math. Soc., 60, 871–896 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    B. Carl and I. Stephani, Entropy, Compactness and Approximation of Operators, Cambridge Univ. Press., Cambridge (1990).CrossRefzbMATHGoogle Scholar
  6. 6.
    F. Cobos and T. Kühn, “Approximation and entropy numbers in Besov spaces of generalized smoothness,” J. Approx. Theory, 160, 56–70 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    R. M. Dudley, “The sizes of compact subsets of Hilbert space and continuity of Gaussian processes,” J. Funct. Anal., 1, 290–330 (1967).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    T. Dunker, T. Kühn, M. A. Lifshits, and W. Linde, “Metric entropy of integration operators and small ball probabilities for the Brownian sheet,” J. Approx. Theory, 101, 63–77 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    V. Goodman, “Characteristics of normal samples,” Ann. Probab., 16, 1281–1290 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Y. Gordon, H. König, and C. Schütt, “Geometric and probabilistic estimates for entropy and approximation numbers of operators,” J. Approx. Theory, 49, 219–239 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    E. Hashorva, M. Lifshits, and O. Seleznjev, “Approximation of a random process with variable smoothness,” in: Mathematical Statistics and Limit Theorems, Springer, Cham (2015), pp. 189–208.Google Scholar
  12. 12.
    H. König, Eigenvalue Distribution of Compact Operators, Birkhäuser, Basel (1986).CrossRefzbMATHGoogle Scholar
  13. 13.
    T. Kühn, “γ-Radonifying operators and entropy ideals,” Math. Nachr., 107, 53–58 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    T. Kühn, “Entropy numbers of general diagonal operators,” Rev. Mat. Complut., 18, 479–491 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    T. Kühn, H.-G. Leopold, W. Sickel, and L. Skrzypczak, “Entropy numbers of embeddings of weighted Besov spaces. II,” Proc. Edinburgh Math. Soc., 49, 331–359 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    T. Kühn and W. Linde, “Optimal series representation of fractional Brownian sheets,” Bernoulli, 8, 669–696 (2002).MathSciNetzbMATHGoogle Scholar
  17. 17.
    J. Kuelbs and W. V. Li, “Metric entropy and the small ball problem for Gaussian measures,” J. Funct. Anal., 116, 133–157 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    W. V. Li and W. Linde, “Approximation, metric entropy and small ball estimates for Gaussian measures,” Ann. Probab., 27, 1556–1578 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    W. Linde and A. Pietsch, “Mappings of Gaussian measures of cylindrical sets in Banach spaces,” Teor. Verojatn. Primen., 19, 472–487 (1974).MathSciNetzbMATHGoogle Scholar
  20. 20.
    V. D. Milman and G. Pisier, “Gaussian processes and mixed volumes,” Ann. Probab., 15, 292–304 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    A. Pajor and N. Tomczak-Jaegermann, “Remarques sur les nombres d’entropie d’un opérateur et de son transposé,” C. R. Acad. Sci. Paris, 301, 743–746 (1985).MathSciNetzbMATHGoogle Scholar
  22. 22.
    A. Pietsch, Eigenvalues and s-Numbers, Cambridge Univ. Press., Cambridge (1987).zbMATHGoogle Scholar
  23. 23.
    G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press., Cambridge (1989).CrossRefzbMATHGoogle Scholar
  24. 24.
    I. Steinwart, “Entropy of C(K)-valued operators,” J. Approx. Theory, 103, 302–328 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    V. N. Sudakov, “Gaussian measures, Cauchy measures and ϵ-entropy,” Soviet Math. Dokl., 10, 310–313 (1969).zbMATHGoogle Scholar
  26. 26.
    M. Talagrand, “Regularity of Gaussian processes,” Acta Math., 159, 99–149 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    M. Talagrand, “The small ball problem for the Brownian sheet,” Ann. Probab., 22, 1331–1354 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    N. Tomczak-Jaegermann, “Dualité des nombres d’entropie pour des opérateurs á valeurs dans un espace de Hilbert,” C.R. Acad. Sci. Paris, 305, 299–301 (1987).MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Universität LeipzigLeipzigGermany
  2. 2.University of DelawareNewarkUSA

Personalised recommendations