Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet

T. Dunker, W. Linde, T. Kühn, M. A. Lifshits

Research output

9 Citations (Scopus)

Abstract

Let Td:L2([0, 1]d)→C([0, 1]d) be the d-dimensional integration operator. We show that its Kolmogorov and entropy numbers decrease with order at least k-1(logk)d-1/2. From this we derive that the small ball probabilities of the Brownian sheet on [0, 1]d under the C([0, 1]d)-norm can be estimated from below by exp(-Cε-2logε2d-1), which improves the best known lower bounds considerably. We also get similar results with respect to certain Orlicz norms.

Original languageEnglish
Pages (from-to)63-77
Number of pages15
JournalJournal of Approximation Theory
Volume101
Issue number1
DOIs
Publication statusPublished - 1 Nov 1999

Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Mathematics(all)
  • Applied Mathematics

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