Hi I have a problem with quadratic splines, I am supposed to find $ S_1 $ and $S_2$ that interpolates the following points $S(-1)=0$, $S(0)=1$, $S(1)=2$, and at the same time we want to find $S$ such that $\int_{-1}^1 \! (S(x))^2 \, \mathrm{d}x$ is minimal. The answer is on the form
$S_1(x)=a_1 x^2 +b_1x +c_1$ on $[-1,0]$ and
$S_2(x)=a_2 x^2 +b_2x +c_2$ on $[0,1]$
my answer:
I use the data points and find that $a_1=-a_2$, $b_1=b_2$ and $c_1=c_2=1$, but I have no idea how to use minimize $\int_{-1}^1 \! (S(x))^2 \, \mathrm{d}x$ , can I divide it up?
$$\min \int_{-1}^1 \! (S(x))^2 \, \mathrm{d}x=\min (\int_{-1}^0 \! (S_1(x))^2 \, \mathrm{d}x + \int_0^1 \! (S_2(x))^2 \, \mathrm{d}x )$$
I know I should get an expression and probably set the derivative to zero but I just don't know how to attack the minimizing integral since the function has two parts. Help greatly appreciated.