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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Artstein, V. D. Milman, and S. Z. Szarek, “Duality of metric entropy,” Ann. Math., 159, 1313–1328 (2004).
D. Bilyk, M. T. Lacey, and A. Vagharshakyan, “On the small ball inequality in all dimensions,” J. Funct. Anal., 254, 2470–2502 (2008).
B. Carl, “Inequalities of Bernstein–Jackson-type and the degree of compactness of operators in Banach spaces,” Ann. Inst. Fourier, 35, 79–118 (1985).
B. Carl, I. Kyrezi, and A. Pajor, “Metric entropy of convex hulls in Banach spaces,” J. London Math. Soc., 60, 871–896 (1999).
B. Carl and I. Stephani, Entropy, Compactness and Approximation of Operators, Cambridge Univ. Press., Cambridge (1990).
F. Cobos and T. Kühn, “Approximation and entropy numbers in Besov spaces of generalized smoothness,” J. Approx. Theory, 160, 56–70 (2009).
R. M. Dudley, “The sizes of compact subsets of Hilbert space and continuity of Gaussian processes,” J. Funct. Anal., 1, 290–330 (1967).
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).
V. Goodman, “Characteristics of normal samples,” Ann. Probab., 16, 1281–1290 (1988).
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).
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.
H. König, Eigenvalue Distribution of Compact Operators, Birkhäuser, Basel (1986).
T. Kühn, “γ-Radonifying operators and entropy ideals,” Math. Nachr., 107, 53–58 (1982).
T. Kühn, “Entropy numbers of general diagonal operators,” Rev. Mat. Complut., 18, 479–491 (2005).
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).
T. Kühn and W. Linde, “Optimal series representation of fractional Brownian sheets,” Bernoulli, 8, 669–696 (2002).
J. Kuelbs and W. V. Li, “Metric entropy and the small ball problem for Gaussian measures,” J. Funct. Anal., 116, 133–157 (1993).
W. V. Li and W. Linde, “Approximation, metric entropy and small ball estimates for Gaussian measures,” Ann. Probab., 27, 1556–1578 (1999).
W. Linde and A. Pietsch, “Mappings of Gaussian measures of cylindrical sets in Banach spaces,” Teor. Verojatn. Primen., 19, 472–487 (1974).
V. D. Milman and G. Pisier, “Gaussian processes and mixed volumes,” Ann. Probab., 15, 292–304 (1987).
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).
A. Pietsch, Eigenvalues and s-Numbers, Cambridge Univ. Press., Cambridge (1987).
G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press., Cambridge (1989).
I. Steinwart, “Entropy of C(K)-valued operators,” J. Approx. Theory, 103, 302–328 (2000).
V. N. Sudakov, “Gaussian measures, Cauchy measures and ϵ-entropy,” Soviet Math. Dokl., 10, 310–313 (1969).
M. Talagrand, “Regularity of Gaussian processes,” Acta Math., 159, 99–149 (1987).
M. Talagrand, “The small ball problem for the Brownian sheet,” Ann. Probab., 22, 1331–1354 (1994).
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Vladimir Nikolaevich Sudakov
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 457, 2017, pp. 194–210.
Rights and permissions
About this article
Cite this article
Kühn, T., Linde, W. Gaussian Approximation Numbers and Metric Entropy. J Math Sci 238, 471–483 (2019). https://doi.org/10.1007/s10958-019-04251-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-019-04251-8