Hello,

i try to solve this problem:

Let $\displaystyle a=x_0 < x_1 <...<x_n =b$ be a partition of [a,b]. and let $\displaystyle V={v \in C^0[a,b] : v_{|[a,b]} \in C^4[x_i-1,x_i]$ with $\displaystyle v(x_i)=y_i $for some $\displaystyle y_i \in \mathbb{R}$. Show that every solution of $\displaystyle argmin_{v \in V} \int |v'(x)|^2 dx$ must be a spline of degree 1.

Do you have a idea, how i can solve this?

It is obvious, that there is a 1 Spline in V (the only one).

Now we have to show that any other element in V is greater than this 1-spline. But how can i do this?

Regards