Metric entropy of convex hulls in type $p$ spaces—The critical case
نویسندگان
چکیده
منابع مشابه
METRIC ENTROPY OF CONVEX HULLS IN TYPE p SPACES—THE CRITICAL CASE
Given a precompact subset A of a type p Banach space E, where p ∈ (1, 2], we prove that for every β ∈ [0, 1) and all n ∈ N sup k≤n k ′ (log k)ek(acoA) ≤ c sup k≤n k ′ (log k)ek(A) holds, where acoA is the absolutely convex hull of A and ek(.) denotes the kth dyadic entropy number. With this inequality we show in particular that for given A and β ∈ (−∞, 1) with en(A) ≤ n−1/p ′ (logn)−β for all n...
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Let A be a subset of a type p Banach space E, 1 < p ≤ 2, such that its entropy numbers satisfy ( εn(A) ) n ∈ `q,s for some q, s ∈ (0,∞). We show ( en(acoA) ) n ∈ `r,s for the dyadic entropy numbers of the absolutely convex hull acoA 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 ∈ `q,s implies ( en(acoA) ) n ∈ `p′,s for al...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 2001
ISSN: 0002-9939,1088-6826
DOI: 10.1090/s0002-9939-01-06256-6