Functional data can be viewed as high dimensional data (infinite dimensional data in some cases) in the sense that each functional object consists of a large number (among infinitely many) measurements. In dependent case, unavoidable serially dependence, interpreted as between curve dynamics, suggests to study functional random processes assuming some stationarity property. Periodically correlated processes, as a natural generalization of stationary processes, play a crucial role in describing random phenomenon which exhibit some periodic behavior. The pollution, traffic, temperature, and financial time series, for example, suggest weekly, monthly, etc. behavior. In this study, we develop the functional version of the spectral representation of random time series, and also investigate autoregressive equations with functional solutions concerning some periodic behavior. \\\\ In more detail, first part of our study prepares a spectral representation of periodically correlated processes based on a method developing functional principal components. In this way, we provide an inversion formula to reconstruct the original data with some optimality property. We also implement our approach on a real data set employing a custom build \\verb|R| package called \\verb|pcdpca|. This practical implementation as well as two simulation studies show our method outperforms the existing methods. \\\\ In the second part, we present necessary and sufficient condition for existing a unique strictly periodically correlated solution to the autoregressive equation of order one, without moment conditions. The moving average representation and limiting distribution of the solution are also investigated. Moreover, we provide condition under which the autoregressive equation of order one admits a weakly periodically solution. The condition is simpler than what is stated in previous studies. Finally, we extend our results to case of the autoregressive equatio with finite order.
