Quantifying the cost of simultaneous non-parametric approximation of several samples

Given $k$ samples $(t_{ij},y(t_{ij}))_{j=1}^{n_i}\,\, t_{ij} \in [0,\,1], i=1,\ldots,\,k$ we consider the problem of finding a function $f_n$, if any, which is simultaneously an adequate approximation for all $k$ samples. Possible measures of cost are the number of local extremes, the total variation of $f_n$ and its derivatives and the supremum norm of derivatives of $f_n$. The costs of a joint approximation can be comparedwith the costs of approximating each sample separately. The concept of approximation used is based on the means of residuals $r_{ij}=y_{ij}-f_n(t_{ij})$ over a family of subsets ${\cal I}_n$ of intervals $I$ of $[0,\,1].$.