Kolmogorov numbers of Riemann-Liouville operators over small sets and applications to Gaussian processes

  • Authors:

  • Werner Linde

  • Affiliations:

  • Faculty of Mathematics and Computer Sciences, Friedrich-Schiller-Universität Jena, Ernst Abbe Platz 2, 07743 Jena, Germany

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2004

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Abstract

We investigate compactness properties of the Riemann-Liouville operator Rx of fractional integration when regarded as operator from L2[0,1] into C(K), the space of continuous functions over a compact subset K in [0,1], Of special interest are small sets K, i.e. those possessing Lebesgue measure zero (e.g. fractal sets). We prove upper estimates for the Kolmogorov numbers of Rx against certain entropy numbers of K. Under some regularity assumption about the entropy of K these estimates turn out to be two-sided. By standard methods the results are also valid for the (dyadic) entropy numbers of Rx. Finally, we apply these estimates for the investigation of the small ball behavior of certain Gaussian stochastic processes, as e.g. fractional Brownian motion or Riemann-Liouville processes, indexed by small (fractal) sets.