Fourier analysis of stationary time series in function space

@article{Panaretos2013FourierAO,
  title={Fourier analysis of stationary time series in function space},
  author={Victor M. Panaretos and Shahin Tavakoli},
  journal={Annals of Statistics},
  year={2013},
  volume={41},
  pages={568-603}
}
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… 

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