Consistently recovering the signal from noisy functional data

In practice most functional data cannot be recorded on a continuum, but rather at discrete time points. It is also quite common that these measurements come with an additive error, which one would like eliminate for the statistical analysis. When each datum are taken same grid, underlying signal-plus-noise model can viewed as factor model. The signals refer to components of model, noise related idiosyncratic components. We formulate framework allows consistently recover signal by PCA based estimation scheme. Our theoretical results hold under mild conditions, in particular we do not require specific smoothness assumptions curves and allow certain degree autocorrelation noise.