Assume we have a dataset containing discretized functional data observed at points t₁, t₂,,tp.In other words, the dataset contains N samples, and each sample contains P noisy observations of a function f(t). That is, the ith observation in the th sample is f(ti) + Cij, where eij is random error. Show that when fitting cubic splines to estimate the mean function, the following three approaches are equivalent: first take the average of all the samples and then fit cubic splines to the sample mean; fit splines to each sample then take the average of the fitted curves; • fit splines to each sample, take the average of the splines coefficients, and reconstruct the mean function by multiplying the average coefficients by the spline bases.