i try to solve this problem:
Let be a partition of [a,b]. and let with for some . Show that every solution of 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?
The only spline of degree one that interpolates the points is in fact the piecewise linear interpolant. Let's assume that the data points are taken from some function such that and denote the piecewise linear interpolant by . Now, let be some other function that interpolates the data points, and define . Then,
If you can show that the second term on the right, i.e. is zero, then you may conclude that
In other words, now you must show that and the degree zero spline are orthogonal. This is quite easy to show:
First, we have divided the interval into parts , then we use integration by parts
Since 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 , so we may write . We find that
This last part is obviously zero because both and satisfy the same interpolation conditions at the points .
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.