  We are interested to detect periodic signals in Hilbert space valued time
series when the length of the period is unknown. A natural test statistic is
the maximum Hilbert-Schmidt norm of the periodogram operator over all
fundamental frequencies. In this paper we analyze the asymptotic distribution
of this test statistic. We consider the case where the noise variables are
independent and then generalize our results to functional linear processes.
Details for implementing the test are provided for the class of functional
autoregressive processes. We illustrate the usefulness of our approach by
examining air quality data from Graz, Austria. The accuracy of the asymptotic
theory in finite samples is evaluated in a simulation experiment.
