论文信息 - Entropy of convex hulls--some Lorentz norm results

Entropy of convex hulls--some Lorentz norm results

Let A be a subset of a type p Banach space E, 1 < p ≤ 2, such that its entropy numbers satisfy, (en(A))n ∈ lq,s for some q,s ∈ (0, ∞). We show (en(aco A))n ∈ lr,s for the dyadic entropy numbers of the absolutely convex hull aco A of A, where r is defined by 1/r = 1/p' + 1/q. Furthermore, we show for slowly decreasing entropy numbers that (en(A))n ∈ lq,s implies (en(aco A))n ∈ lp',s for all 0 < s < ∞ and q defined by 1/q = 1/p' + 1/s.

Ingo Steinwart

[1] B. Carl,et al. Metric Entropy of Convex Hulls in Banach Spaces , 1999 .

[2] Wenbo V. Li,et al. Metric entropy of convex hulls in Hilbert spaces , 2000 .

[3] Ingo Steinwart. Entropy of C(K)-Valued Operators , 2000 .

[4] A. Pietsch. Eigenvalues and S-Numbers , 1987 .

[5] G. Pisier,et al. Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach , 1976 .

[6] G. Pisier. Remarques sur un résultat non publié de B. Maurey , 1981 .

[7] Fuchang Gao. Metric entropy of convex hulls , 2001 .

[8] B. Carl. Metric Entropy of Convex Hulls in Hilbert Spaces , 1997 .

[9] B. Carl,et al. s-Numbers of integral operators with Hölder continuous kernels over metric compacta , 1988 .

[10] Ingo Steinwart,et al. Metric entropy of convex hulls in type p spaces-the critical case , 2001 .

[11] Entropy of C(K)-valued operators and some applications , 2000 .

[12] B. Carl,et al. Entropy, Compactness and the Approximation of Operators , 1990 .

[13] R. Dudley. Universal Donsker Classes and Metric Entropy , 1987 .

[14] On the entropy of the convex hull of finite sets , 1999 .

[15] Jon A. Wellner,et al. Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .