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Ingo Steinwart (2004)
Entropy of convex hulls--some Lorentz norm resultsJ. Approx. Theory, 128
(1984)
On widths of the Euclidean ball (Russian)
B. Carl (1981)
Entropy numbers, s-numbers, and eigenvalue problemsJournal of Functional Analysis, 41
J. Graaf (1998)
Book review: Function spaces, entropy numbers and differential operators, 41
J. Creutzig, Ingo Steinwart (2001)
Metric entropy of convex hulls in type p spaces-the critical case, 130
R. Dudley (1973)
Sample Functions of the Gaussian ProcessAnnals of Probability, 1
B. Carl, A. Pajor (1988)
Gelfand numbers of operators with values in a Hilbert spaceInventiones mathematicae, 94
J. Kuelbs (1978)
Probability on Banach spaces
(1989)
Volume Inequalities in the Geometry of Banach Spaces
R. Dudley (1967)
The Sizes of Compact Subsets of Hilbert Space and Continuity of Gaussian ProcessesJournal of Functional Analysis, 1
B. Carl, S. Heinrich, T. Kühn (1988)
s-Numbers of integral operators with Hölder continuous kernels over metric compactaJournal of Functional Analysis, 81
B. Carl, D. Edmunds (2001)
Entropy of C(Z)-Valued Operators and Diverse ApplicationsJournal of Inequalities and Applications, 2001
S. Artstein, V. Milman, S. Szarek (2004)
Duality of metric entropyAnnals of Mathematics, 159
Wenbo Li, W. Linde (2000)
Metric entropy of convex hulls in Hilbert spacesStudia Mathematica, 139
Fuchang Gao (2001)
Metric entropy of convex hullsIsrael Journal of Mathematics, 123
A.W. Vaart, J.A. Wellner (1996)
Weak convergence and empirical processes
B. Carl, A. Hinrichs, A. Pajor (1997)
Gelfand numbers and metric entropy of convex hulls in Hilbert spacesPositivity, 17
C. Schütt (1984)
Entropy numbers of diagonal operators between symmetric Banach spacesJournal of Approximation Theory, 40
R. Dudley (1987)
Universal Donsker Classes and Metric EntropyAnnals of Probability, 15
(1999)
Approximation, metric entropy and small ball estimates for Gaussian measures
F. Gao (2004)
Entropy of convex hulls in Hilbert spacesBull. Lond. Math. Soc., 36
A. Pietsch (1987)
Eigenvalues and S-Numbers
Ingo Steinwart (2000)
Entropy of C(K)-Valued OperatorsJournal of Approximation Theory, 103
M. Talagrand, Michel Talagrand (1993)
New Gaussian estimates for enlarged ballsGeometric & Functional Analysis GAFA, 3
M. Talagrand (1987)
Regularity of gaussian processesActa Mathematica, 159
B. Carl (1982)
On a characterization of operators from lq into a Banach space of type p with some applications to eigenvalue problemsJournal of Functional Analysis, 48
B. Carl (1985)
Inequalities of Bernstein–Jackson type and the degree of compactness of operators in Banach spacesAnn. Inst. Fourier, 35
B. Carl (1985)
Inequalities of Bernstein-Jackson-type and the degree of compactness of operators in Banach spacesAnnales de l'Institut Fourier, 35
B. Carl, I. Kyrézi, A. Pajor (1999)
Metric Entropy of Convex Hulls in Banach SpacesJournal of the London Mathematical Society, 60
B. Carl (1982)
On a characterization of operators from l q into a Banach space of type p with some applications to eigenvalue problemsJ. Funct. Anal., 48
K. Ball, A. Pajor, P. Muller, W. Schachermayer (1991)
The entropy of convex bodies with ‘few’ extreme points
José González-Barrios, R. Dudley (1993)
Metric entropy conditions for an operator to be of trace class, 118
B. Carl, H. Triebel (1980)
Inequalities between eigenvalues, entropy numbers, and related quantities of compact operators in Banach spacesMathematische Annalen, 251
B. Carl (1997)
Metric entropy of convex hulls in Hilbert spacesBull. Lond. Math. Soc., 29
B. Carl, Irmtraud Stephani (1990)
Entropy, Compactness and the Approximation of Operators
We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets $${K\subset H}$$ of a Hilbert space H by the metric entropy of the set K where the covering numbers $${N(K, \varepsilon)}$$ of K by $${\varepsilon}$$ -balls of H satisfy the Lorentz condition $$ \int\limits_{0}^{\infty} \left(\log N(K,\varepsilon) \right)^{r/s}\, d\varepsilon^{s} < \infty $$ for some fixed $${0 < r, s \le \infty}$$ with the usual modifications in the cases r = ∞, 0 < s < ∞ and 0 < r < ∞, s = ∞. The integral here is an improper Stieltjes integral. Moreover, we obtain optimal estimates of Gelfand numbers of absolutely convex hulls if the metric entropy satisfies the entropy condition $$\sup_{\varepsilon >0 }\varepsilon \left(\log N(K,\varepsilon) \right)^{1/r}\left(\log(2+\log N(K,\varepsilon))\right)^\beta < \infty$$ for some fixed 0 < r < ∞, −∞ < β < ∞. Using inequalities between Gelfand and entropy numbers we also get optimal estimates of the metric entropy of the absolutely convex hull cov(K). As an interesting feature of the estimates, a sudden jump of the asymptotic behavior of Gelfand numbers as well as of the metric entropy of absolutely convex hulls occurs for fixed s if the parameter r crosses the point r = 2 and, if r = 2 is fixed, if the parameter β crosses the point β = 1. The results established in Hilbert spaces extend and recover corresponding results of several authors.
Positivity – Springer Journals
Published: Jan 28, 2012
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