Series Representations of Fractional Gaussian Processes by Trigonometric and Haar Systems

The aim of the present paper is to investigate series representations of the Riemann–Liouville process R, α > 1/2, generated by classical orthonormal bases in L2[0,1]. Those bases are, for example, the trigonometric or the Haar system. We prove that the representation of R via the trigonometric system possesses the optimal convergence rate if and only if 1/2 < α ≤ 2. For the Haar system we have an optimal approximation rate if 1/2 < α < 3/2 while for α > 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 α > 3/2 and α= 3/2. However, in this latter case the question whether or not the series representation is optimal remains open. 1 .