Series Representations of Fractional Gaussian Processes by Trigonometric and Haar Systems

摘要

The aim of the present paper is to investigate series representations of the Riemann-Liouville process $R^lpha$, $lpha >1/2$, generated by classical orthonormal bases in $L_2[0,1]$. Those bases are, for example, the trigonometric or the Haar system. We prove that the representation of $R^lpha$ via the trigonometric system possesses the optimal convergence rate if and only if $1/2 3/2$ a representation via the Haar system is not optimal. Estimates for the rate of convergence of the Haar series are given in the cases $lpha > 3/2$ and $lpha = 3/2$. However, in this latter case the question whether or not the series representation is optimal remains open. Recently M. A. Lifshits answered this question (cf. [13]). Using a different approach he could show that in the case $lpha = 3/2$ a representation of the Riemann-Liouville process via the Haar system is also not optimal.