Geodesic
Gaussian approximation numbers and metric entropy
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 25, Tome 457 (2017), pp. 194-210 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The aim of this paper is to survey properties of Gaussian approximation numbers. We state the basic relations between these numbers and 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 a final section we apply the relations between Gaussian and other $s$-numbers to the $d$-dimensional integration operator defined on $L_2[0,1]^d$. 
@article{ZNSL_2017_457_a10,
     author = {T. K\"uhn and W. Linde},
     title = {Gaussian approximation numbers and metric entropy},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {194--210},
     year = {2017},
     volume = {457},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_457_a10/}
}
TY  - JOUR
AU  - T. Kühn
AU  - W. Linde
TI  - Gaussian approximation numbers and metric entropy
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2017
SP  - 194
EP  - 210
VL  - 457
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2017_457_a10/
LA  - en
ID  - ZNSL_2017_457_a10
ER  - 
%0 Journal Article
%A T. Kühn
%A W. Linde
%T Gaussian approximation numbers and metric entropy
%J Zapiski Nauchnykh Seminarov POMI
%D 2017
%P 194-210
%V 457
%U http://geodesic.mathdoc.fr/item/ZNSL_2017_457_a10/
%G en
%F ZNSL_2017_457_a10
T. Kühn; W. Linde. Gaussian approximation numbers and metric entropy. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 25, Tome 457 (2017), pp. 194-210. http://geodesic.mathdoc.fr/item/ZNSL_2017_457_a10/

[1] S. Artstein, V. D. Milman, S. Z. Szarek, “Duality metric entropy”, Ann. of Math., 159 (2004), 1313–1328 | DOI | MR | Zbl

[2] D. Bilyk, M. T. Lacey, A. Vagharshakyan, “On the small ball inequality in all dimensions”, J. Funct. Anal., 254 (2008), 2470–2502 | DOI | MR | Zbl

[3] B. Carl, “Inequalities of Bernstein–Jackson-type and the degree of compactness of operators in Banach spaces”, Ann. Inst. Fourier, 35 (1985), 79–118 | DOI | MR | Zbl

[4] B. Carl, I. Kyrezi, A. Pajor, “Metric entropy of convex hulls in Banach spaces”, J. London Math. Soc., 60 (1999), 871–896 | DOI | MR | Zbl

[5] B. Carl, I. Stephani, Entropy, Compactness and Approximation of Operators, Cambridge Univ. Press., Cambridge, 1990 | MR | Zbl

[6] F. Cobos, T. Kühn, “Approximation and entropy numbers in Besov spaces of generalized smoothness”, J. Approx. Theory, 160 (2009), 56–70 | DOI | MR | Zbl

[7] R. M. Dudley, “The sizes of compact subsets of Hilbert space and continuity of Gaussian processes”, J. Funct. Anal., 1 (1967), 290–330 | DOI | MR | Zbl

[8] T. Dunker, T. Kühn, M. A. Lifshits, W. Linde, “Metric entropy of integration operators and small ball probabilities for the Brownian sheet”, J. Approx. Theory, 101 (1999), 63–77 | DOI | MR | Zbl

[9] V. Goodman, “Characteristics of normal samples”, Ann. Probab., 16 (1988), 1281–1290 | DOI | MR | Zbl

[10] Y. Gordon, H. König, C. Schütt, “Geometric and probabilistic estimates for entropy and approximation numbers of operators”, J. Approx. Theory, 49 (1987), 219–239 | DOI | MR | Zbl

[11] E. Hashorva, M. Lifshits, O. Seleznjev, “Approximation of a random process with variable smoothness”, Mathematical statistics and limit theorems, Springer, Cham, 2015, 189–208 | MR | Zbl

[12] H. König, Eigenvalue Distribution of Compact Operators, Birkhäuser, Basel, 1986 | MR | Zbl

[13] T. Kühn, “$\gamma$-Radonifying operators and entropy ideals”, Math. Nachr., 107 (1982), 53–58 | DOI | MR | Zbl

[14] T. Kühn, “Entropy numbers of general diagonal operators”, Rev. Mat. Complut., 18 (2005), 479–491 | DOI | MR | Zbl

[15] T. Kühn, H.-G. Leopold, W. Sickel, L. Skrzypczak, “Entropy numbers of embeddings of weighted Besov space. II”, Proc. Edinburgh Math. Soc., 49 (2006), 331–359 | DOI | MR | Zbl

[16] T. Kühn, W. Linde, “Optimal series representation of fractional Brownian sheets”, Bernoulli, 8 (2002), 669–696 | MR | Zbl

[17] J. Kuelbs, W. V. Li, “Metric entropy and the small ball problem for Gaussian measures”, J. Funct. Anal., 116 (1993), 133–157 | DOI | MR | Zbl

[18] W. V. Li, W. Linde, “Approximation, metric entropy and small ball estimates for Gaussian measures”, Ann. Probab., 27 (1999), 1556–1578 | DOI | MR | Zbl

[19] W. Linde, A. Pietsch, “Mappings of Gaussian measures of cylindrical sets in Banach spaces”, Teor. Verojatnost. i Primenen., 19:3 (1974), 472–487 | MR | Zbl

[20] V. D. Milman, G. Pisier, “Gaussian processes and mixed volumes”, Ann. Probab., 15 (1987), 292–304 | DOI | MR | Zbl

[21] A. Pajor, N. Tomczak-Jaegermann, “Remarques sur les nombres d'entropie d'un opérateur et de son transposé”, C. R. Acad. Sci. Paris, 301 (1985), 743–746 | MR | Zbl

[22] A. Pietsch, Eigenvalues and s-Numbers, Cambridge Univ. Press, Cambridge, 1987 | MR | Zbl

[23] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press, Cambridge, 1989 | MR | Zbl

[24] I. Steinwart, “Entropy of $C(K)$-valued operators”, J. Approx. Theory, 103 (2000), 302–328 | DOI | MR | Zbl

[25] V. N. Sudakov, “Gaussian measures, Cauchy measures and $\epsilon$-entropy”, Soviet Math. Dokl., 10 (1969), 310–313 | Zbl

[26] M. Talagrand, “Regularity of Gaussian processes”, Acta Math., 159 (1987), 99–149 | DOI | MR | Zbl

[27] M. Talagrand, “The small ball problem for the Brownian sheet”, Ann. Probab., 22 (1994), 1331–1354 | DOI | MR | Zbl

[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 (1987), 299–301 | MR | Zbl