Kolmogorov numbers of Riemann-Liouville operators over small sets and applications to Gaussian processes
The author studies the compactnesss properties of the Riemann-Liouville fractional integral operator \(R_\alpha\) from \(L_{2}[0,1]\) into \(C(K)\), where \(K\) is any of certain compact subsets of [0,1]. The degree of compactness of \(R_\alpha\) is measured using the Kolmogorov numbers \(d_ n(K)\) and the entropy numbers \(\varepsilon_ n(K)\) of the underlying set \(K\). A relationship between these two numbers is established and for the sake of completeness, we reproduce one of the results obtained: Suppose that \(\alpha > \frac12\) and let \( K \subset [0,1]\) be a compact set such that for some \(\lambda \geq 1\) \[ \varepsilon_{m}(K) \leq \lambda\cdot \varepsilon_{2m}(K), \quad m\in {\mathbb N}. \] Then it follows that \[ d_ n (R_\alpha : L_ 2 [0,1] \to C(K)) \geq c\cdot m^{-\frac1{2}}\cdot \epsilon_ m (K)^{\alpha-\frac12}, \] where \(c > 0\) only depends on \(\alpha\) and \(\lambda\). Applications of the obtained results to the small ball behaviour of certain Gaussian stochastic processes such as fractional Brownian motion or Riemann-Liouville processes indexed by small (fractal) sets are given.
scientific article; zbMATH DE number 2085994
scientific article