Interest in functional time series has spiked in the recent past with paperscovering both methodology and applications being published at a much increasedpace. This article contributes to the research in this area by proposingstationarity tests for functional time series based on frequency domainmethods. Setting up the tests requires a delicate understanding of periodogram-and spectral density operators that are the functional counterparts ofperiodogram- and spectral density matrices in the multivariate world. Two setsof statistics are proposed. One is based on the eigendecomposition of thespectral density operator, the other on a fixed projection basis. Theirproperties are derived both under the null hypothesis of stationary functionaltime series and under the smooth alternative of locally stationary functionaltime series. The methodology is theoretically justified through asymptoticresults. Evidence from simulation studies and an application to annualtemperature curves suggests that the tests work well in finite samples.
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