Bounds for entropy numbers for some critical operators

Lifshits, M.

doi:10.1090/S0002-9947-2011-05407-3

Bounds for entropy numbers for some critical operators
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Trans. Amer. Math. Soc. 364 (2012), 1797-1813 Request permission

Abstract:

We provide upper bounds for entropy numbers for two types of operators: summation operators on binary trees and integral operators of Volterra type. Our efforts are concentrated on the critical cases where none of the known methods work. Therefore, we develop a method which seems to be completely new and probably merits further applications.

References

Shiri Artstein, Vitali D. Milman, and Stanislaw J. Szarek, Duality of metric entropy in Euclidean space, C. R. Math. Acad. Sci. Paris 337 (2003), no. 11, 711–714 (English, with English and French summaries). MR 2030407, DOI 10.1016/j.crma.2003.09.033
S. Artstein, V. Milman, and S. J. Szarek, Duality of metric entropy, Ann. of Math. (2) 159 (2004), no. 3, 1313–1328. MR 2113023, DOI 10.4007/annals.2004.159.1313
Frank Aurzada and Mikhail Lifshits, Small deviation probability via chaining, Stochastic Process. Appl. 118 (2008), no. 12, 2344–2368. MR 2474354, DOI 10.1016/j.spa.2008.01.005
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
Bernd Carl and Alain Pajor, Gel′fand numbers of operators with values in a Hilbert space, Invent. Math. 94 (1988), no. 3, 479–504. MR 969241, DOI 10.1007/BF01394273
Bernd Carl and Irmtraud Stephani, Entropy, compactness and the approximation of operators, Cambridge Tracts in Mathematics, vol. 98, Cambridge University Press, Cambridge, 1990. MR 1098497, DOI 10.1017/CBO9780511897467
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
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
Fuchang Gao, Metric entropy of convex hulls, Israel J. Math. 123 (2001), 359–364. MR 1835305, DOI 10.1007/BF02784136
M. Lacey, Private communication (2008).
Wenbo V. Li and Werner Linde, Metric entropy of convex hulls in Hilbert spaces, Studia Math. 139 (2000), no. 1, 29–45. MR 1763043
Werner Linde, Nondeterminism of linear operators and lower entropy estimates, J. Fourier Anal. Appl. 14 (2008), no. 4, 568–587. MR 2421577, DOI 10.1007/s00041-008-9023-3
Carsten Schütt, Entropy numbers of diagonal operators between symmetric Banach spaces, J. Approx. Theory 40 (1984), no. 2, 121–128. MR 732693, DOI 10.1016/0021-9045(84)90021-2
Nicole Tomczak-Jaegermann, Dualité des nombres d’entropie pour des opérateurs à valeurs dans un espace de Hilbert, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 7, 299–301 (French, with English summary). MR 910364