
Splines
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 $\displaystyle V={v \in C^0[a,b] : v_{[a,b]} \in C^4[x_i1,x_i]$ 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 1spline. But how can i do this?
Regards

The only spline of degree one that interpolates the points $\displaystyle (x_i,y_i)$ is in fact the piecewise linear interpolant. Let's assume that the data points are taken from some function $\displaystyle f$ such that $\displaystyle y_i = f(x_i)$ and denote the piecewise linear interpolant by $\displaystyle g = If$. Now, let $\displaystyle h \in V$ be some other function that interpolates the data points, and define $\displaystyle e = h  g$. Then,
$\displaystyle \int_a^b (h')^2 dx = \int_a^b (e'+g')^2 dx = \int_a^b (e')^2 dx + 2\int_a^b e'g' dx + \int_a^b (g')^2 dx $
If you can show that the second term on the right, i.e. $\displaystyle \int_a^b e'h' dx$ is zero, then you may conclude that
$\displaystyle \int_a^b (g')^2 dx \leq \int_a^b (h')^2 dx$.
In other words, now you must show that $\displaystyle e'$ and the degree zero spline $\displaystyle g'$ are orthogonal. This is quite easy to show:
$\displaystyle \int_a^b e'g' dx = \sum_{i=1}^{n1} \int_{x_i}^{x_{i+1}} e'g' dx $
First, we have divided the interval $\displaystyle [a,b]$ into parts $\displaystyle [x_i,x_{i+1}]$, then we use integration by parts
$\displaystyle \sum_{i=1}^{n1} \int_{x_i}^{x_{i+1}} e'g' dx = \sum_{i=1}^{n1} \left( eg' \mid_{x_i}^{x_{i+1}}  \int_{x_i}^{x_{i+1}} eg'' dx \right) $
Since $\displaystyle g'$ is a degree zero spline, its derivative is obviously zero, so the integral on the right is zero. Only the sum remains, and now we use the fact that $\displaystyle g=Ih=If$, so we may write $\displaystyle e=hIh$. We find that
$\displaystyle \sum_{i=1}^{n1} e(x)g'(x) \mid_{x_i}^{x_{i+1}} = \sum_{i=1}^{n1} (h(x_{i+1})  g(x_{i+1}))g'(x_{i+1})  (h(x_{i})  g(x_{i}))g'(x_{i}) = 0 $
This last part is obviously zero because both $\displaystyle h$ and $\displaystyle g$ satisfy the same interpolation conditions at the points $\displaystyle x_i$.
I based this on a similar proof in a compendium that was written for a course in splines which I'm taking. Here the proof was about the cubic spline interpolant which minimizes the integral of the second derivative squared. If you're curious, send me a PM and I'll see if I can find a link.