The authors study integration with respect to fractional Brownian motion. Their aim is to characterize the linear span of the fractional Brownian motion , , , in terms of some function space. This means that they study integration of deterministic functions. For it is possible to give a complete characterization of the space as follows: Let
where is the fractional integral operator of order . For put
Then the space is isometric to . For the authors show that it is not possible to obtain such a characterization in terms of a function space.
The authors also study related problems in the spectral domain using the spectral representation of fractional Brownian motion in terms of a complex Gaussian measure. They consider the following function space
They show that the elementary functions are dense in , but this space is not complete, unless . In addition, they show that , the inclusion is strict, when , and where the space for is defined as