The study of functional data analysis is motivated by their applications in various fields of statistical estimation and statistical inverse problems.

In this thesis we propose a functional Hodrick-Prescott filter. This filter is applied to functional data which take values in an infinite dimensional separable Hilbert space.  The filter depends on a smoothing parameter. In this study we characterize the associated optimal smoothing parameter when the underlying distribution of the data is Gaussian. Furthermore we extend this characterization to the case when the underlying distribution of the data is white noise.