  We develop the basic building blocks of a frequency domain framework for
drawing statistical inferences on the second-order structure of a stationary
sequence of functional data. The key element in such a context is the spectral
density operator, which generalises the notion of a spectral density matrix to
the functional setting, and characterises the second-order dynamics of the
process. Our main tool is the functional Discrete Fourier Transform (fDFT). We
derive an asymptotic Gaussian representation of the fDFT, thus allowing the
transformation of the original collection of dependent random functions into a
collection of approximately independent complex-valued Gaussian random
functions. Our results are then employed in order to construct estimators of
the spectral density operator based on smoothed versions of the periodogram
kernel, the functional generalisation of the periodogram matrix. The
consistency and asymptotic law of these estimators are studied in detail. As
immediate consequences, we obtain central limit theorems for the mean and the
long-run covariance operator of a stationary functional time series. Our
results do not depend on structural modelling assumptions, but only functional
versions of classical cumulant mixing conditions, and are shown to be stable
under discrete observation of the individual curves.
