I met the following problem when working on non-parametric statistics.
We have $(y_i,t_i)$ a pair of points in $\mathbb{R^{2}}$. We want to find a smooth function that allows the best fit to the $y_i$. For this, we consider the space $W^{m}([a,b])=\{ f : [a,b]\to\mathbb{R} : \int_{a}^{b} (f^{m}(t))^{2}dt<\infty \}$ where the integral should be viewed as a measure of roughness. In order to answer the following problem, my textbook put the following minimization problem : $arg min_{f\in W^{m}([a,b])}\sum_{i=1}^{n}(y_i -f(t_i))^2 + \lambda\int_{a}^{b} (f^{m}(t))^{2}dt$
However, I am not able to see where it comes from and even less to solve it. Does anyone knows at first where it comes from ?
Thank you a lot