Exercise 13.3: Cubic Smoothing Splines
From Example 12.7, we know that $k(x, y)$ is a kernel for $\Hb_0^2[0, 1]$, the collection of functions $f\colon [0, 1] \to \R$ that are $\alpha$-times differentiable almost everywhere with (1) $f^{\prime\prime}$ Lebesgue-integrable and (2) the constraint that $f(0) = f’(0) = 0$. The example also shows that \begin{equation} \lra{f, g}_k = \int_0^1 f^{\prime\prime}(x)g^{\prime\prime}(x) \isd x. \end{equation} Due to the requirement $f(0) = f’(0) = 0$, the only linear function that $\Hb_0^2[0, 1]$ contains is the zero function. Therefore, every function in $f \in \Hb^2[0, 1]$ can uniquely be additively decomposed in terms of a linear function and a function $f_0 \in \Hb_0^2[0, 1]$. Moreover, \begin{equation} \int_0^1 (f^{\prime\prime}(x))^2 \isd x = \norm{f_0}_{k}^2. \end{equation} The parts therefore follow from the usual representer argument.