Strongly and weakly harmonizable stochastic processes of H-valued random variables

Let H be a Hilbert space and ([Omega], , [mu]) be a probability measure space. Consider the Hilbert space L02([Omega]; H) consisting of all H-valued strong random variables on [Omega] with zero mean which are square integrable with respect to [mu]. We study L02([Omega]; H)-valued processes over the real line R. Strong and weak harmonizabilities are defined for such processes. It is shown that, as in the scalar valued case, every weakly harmonizable process is approximated pointwisely on R by a sequence of strongly harmonizable processes. To prove this we obtain a series representation of a continuous process.