I am given a the following problem set:
Let $s$ be a natural cubic spline that interpolates a function $f \in \mathcal{C}^2 ([a,b])$ at points $a = x_0 < x_1 < \ldots < x_n =b$ with values $ \ f(x_j) = y_j$ for $j \in \{0, \ldots, n\}$. Moreover, for real-valued functions $ g \in \mathcal{C}^0 ([a,b])$ we define the $L^2$-norm by: $$\Vert g\Vert_{L^2} := \left( \int_{a}^{b} |g(x)|^2 dx \right)^{\frac{1}{2}}$$
show that for $j \in \{1, \ldots,n\}$ on each interval $ [x_{j-1},x_j ]$ we have $$ \int_{x_{j-1}}^{x_j} (f^{\prime \prime}(x) - s^{\prime \prime}(x))s^{\prime \prime}(x)dx = (f^{ \prime}(x)- s^{ \prime}(x))s^{\prime \prime}(x)\vert_{x_{j-1}}^{x_j}$$
my problem is to understand exactly what im supposed to do. I can see that this will somehow have to do something with the integration by parts but i need help to get there. thank you
$$\int (f"-s")s"dx = (f'-s')s"-[(f-s)s'''-\int (f-s)s''''dx$$ This is what you get after using integration by parts twice, the last two terms cancel out because $f-s=0$ because your spline equals the function in the nodes.