Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

, Volume 17 (1) – Jan 28, 2012
33 pages

/lp/springer_journal/gelfand-numbers-and-metric-entropy-of-convex-hulls-in-hilbert-spaces-ThMdMmw5Xf
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-012-0157-9
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

PositivitySpringer Journals

Published: Jan 28, 2012

References

• Metric entropy of convex hulls in Hilbert spaces
Carl, B.
• Gelfand numbers and metric entropy of convex hulls in Hilbert spaces
Carl, B.; Edmunds, D.E.

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