Ex. 5.11

Ex. 5.11

Prove that for a smoothing spline the null space of \(\bb{K}\) is spanned by functions linear in \(X\).

Soln. 5.11

First recall the definition of \(\bm{\Omega}_N\) is

\[\begin{equation} \{\bm{\Omega}_N\}_{jk} = \int N''_j(t)N''_k(t)dt.\non \end{equation}\]

Since \(N_1 =1\) and \(N_2 = x\), both have vanished second order derivative, thus we have

\[\begin{equation} \bm{\Omega}_N = \begin{pmatrix} 0& 0&\cdots&0\\ 0& 0&\cdots&0\\ \vdots & \vdots &\ddots &\vdots\\ 0& 0&\cdots& \int N''_N(t)N''Nk(t)dt \end{pmatrix}.\non \end{equation}\]

Thus, for any \(x^T=(c_1, c_2x, 0,...,0)\) where \(c_1\) and \(c_2\) are constants, it's easy to show that

\[\begin{equation} \bm{\Omega}_N\bb{N}^{-1}x = \bb{0}\non \end{equation}\]

so that

\[\begin{equation} \bb{K}x=\bb{0}.\non \end{equation}\]