منابع مشابه
Metric Entropy of Convex Hulls
Let T be a precompact subset of a Hilbert space. The metric entropy of the convex hull of T is estimated in terms of the metric entropy of T , when the latter is of order α = 2. The estimate is best possible. Thus, it answers a question left open in [LL] and [CKP]. 0.
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We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets K ⊂ H of a Hilbert space H by the metric entropy of the set K where the covering numbers N(K, ") of K by "-balls of H satisfy the Lorentz condition ∫ ∞ 0 ( log2N(K, ") )r/s d" <∞ for some fixed 0 < r, s ≤ ∞ with the usual modifications in the cases r = ∞, 0 < s < ∞ and 0 <...
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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: Israel Journal of Mathematics
سال: 2001
ISSN: 0021-2172,1565-8511
DOI: 10.1007/bf02784136