Smooth functions and local extreme values

Given a sample of

observations $y_1,\dots,y_n$ at time points $t_1,\dots,t_n$ we consider the problem of specifying a function $\tilde f$ such that $\tilde f$

is smooth,
fits the data in the sense that the residuals $y_i-\tilde f(t_i)$ satisfy the multiresolution criterion

$\displaystyle \left\vert \frac{1}{\sqrt{k-j+1}}\sum_{i=j}^k y_i-\tilde f(t_i) \right\vert < \sqrt{2\log(n)}\sigma\quad 1\leq j\leq k\leq n,$ (1)
is as simple as possible so that $\tilde f$ exhibits the minimum number of local extreme values.

We analyse in particular a fast method which is based on minimising

$\displaystyle \sum_{i=1}^n(y_i- f(t_i))^2+\sum_{i=1}^{n-1}\lambda_i \sqrt{(f_{i+1}-f_i)^2+(t_{i+1}-t_i)^2}$

where the $\lambda_i$ are chosen automatically. The new method can also be applied to density estimation.