Metric entropy of $q$-hulls in Banach spaces of type-$p$
HTML articles powered by AMS MathViewer
- by James Cockreham, Fuchang Gao and Yuhong Yang PDF
- Proc. Amer. Math. Soc. 145 (2017), 5205-5214 Request permission
Abstract:
Optimal upper bound estimates for metric entropy of $l_q$-hulls, $0<q\le 1$, are obtained for precompact sets in Banach spaces of type-$p$.References
- Keith Ball and Alain Pajor, The entropy of convex bodies with “few” extreme points, Geometry of Banach spaces (Strobl, 1989) London Math. Soc. Lecture Note Ser., vol. 158, Cambridge Univ. Press, Cambridge, 1990, pp. 25–32. MR 1110183
- Bernd Carl, Metric entropy of convex hulls in Hilbert spaces, Bull. London Math. Soc. 29 (1997), no. 4, 452–458. MR 1446564, DOI 10.1112/S0024609397003044
- Bernd Carl, Ioanna Kyrezi, and Alain Pajor, Metric entropy of convex hulls in Banach spaces, J. London Math. Soc. (2) 60 (1999), no. 3, 871–896. MR 1753820, DOI 10.1112/S0024610799008005
- Jakob Creutzig and Ingo Steinwart, Metric entropy of convex hulls in type $p$ spaces—the critical case, Proc. Amer. Math. Soc. 130 (2002), no. 3, 733–743. MR 1866028, DOI 10.1090/S0002-9939-01-06256-6
- R. M. Dudley, The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Functional Analysis 1 (1967), 290–330. MR 0220340, DOI 10.1016/0022-1236(67)90017-1
- D. E. Edmunds and H. Triebel, Function spaces, entropy numbers, differential operators, Cambridge Tracts in Mathematics, vol. 120, Cambridge University Press, Cambridge, 1996. MR 1410258, DOI 10.1017/CBO9780511662201
- Simon Foucart, Alain Pajor, Holger Rauhut, and Tino Ullrich, The Gelfand widths of $\ell _p$-balls for $0<p\leq 1$, J. Complexity 26 (2010), no. 6, 629–640. MR 2735423, DOI 10.1016/j.jco.2010.04.004
- Fuchang Gao, Metric entropy of convex hulls, Israel J. Math. 123 (2001), 359–364. MR 1835305, DOI 10.1007/BF02784136
- Fuchang Gao, Entropy of absolute convex hulls in Hilbert spaces, Bull. London Math. Soc. 36 (2004), no. 4, 460–468. MR 2069008, DOI 10.1112/S0024609304003121
- Fuchang Gao, Optimality of CKP-inequality in the critical case, Proc. Amer. Math. Soc. 142 (2014), no. 3, 909–914. MR 3148525, DOI 10.1090/S0002-9939-2013-11825-3
- Fuchang Gao, Ching-Kang Ing, and Yuhong Yang, Metric entropy and sparse linear approximation of $\ell _q$-hulls for $0<q\leq 1$, J. Approx. Theory 166 (2013), 42–55. MR 3003947, DOI 10.1016/j.jat.2012.10.002
- Fuchang Gao, Wenbo V. Li, and Jon A. Wellner, How many Laplace transforms of probability measures are there?, Proc. Amer. Math. Soc. 138 (2010), no. 12, 4331–4344. MR 2680059, DOI 10.1090/S0002-9939-2010-10448-3
- Evarist Giné and Joel Zinn, Some limit theorems for empirical processes, Ann. Probab. 12 (1984), no. 4, 929–998. With discussion. MR 757767
- James Kuelbs and Wenbo V. Li, Metric entropy and the small ball problem for Gaussian measures, J. Funct. Anal. 116 (1993), no. 1, 133–157. MR 1237989, DOI 10.1006/jfan.1993.1107
- G. Pisier, Remarques sur un résultat non publié de B. Maurey, Seminar on Functional Analysis, 1980–1981, École Polytech., Palaiseau, 1981, pp. Exp. No. V, 13 (French). MR 659306
- S. A. Smoljak, $\varepsilon$-entropy of the classes $E_{s^{\alpha ,k}}(B)$ and $W_{s^{\alpha }}\,(B)$ in the metric of $L_{2}$, Soviet Math. Dokl. 1 (1960), 192–196. MR 0123130
- A. F. Timan, Theory of approximation of functions of a real variable, A Pergamon Press Book, The Macmillan Company, New York, 1963. Translated from the Russian by J. Berry; English translation edited and editorial preface by J. Cossar. MR 0192238
Additional Information
- James Cockreham
- Affiliation: Department of Mathematics, University of Idaho, Moscow, Idaho 83844-1103
- Email: jmcockreham@alaska.edu
- Fuchang Gao
- Affiliation: Department of Mathematics, University of Idaho, Moscow, Idaho 83844-1103
- MR Author ID: 290983
- Email: fuchang@uidaho.edu
- Yuhong Yang
- Affiliation: School of Statistics, University of Minnesota, Minneapolis, Minnesota 55455
- MR Author ID: 627909
- Email: yyang@stat.umn.edu
- Received by editor(s): November 2, 2015
- Received by editor(s) in revised form: December 19, 2016
- Published electronically: June 22, 2017
- Additional Notes: The research of the second author was partially supported by a grant from the Simons Foundation, #246211.
The research of the third author was partially supported by the National Natural Science Foundation of China, #61572109 - Communicated by: Thomas Schlumprecht
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 5205-5214
- MSC (2010): Primary 47B06, 52A27, 41A46
- DOI: https://doi.org/10.1090/proc/13627
- MathSciNet review: 3717949