Assuming that a B-spline is defined for some then the B-spline is defined by:
How can I, through induction, show that:
Suggestions would be greatly appreciated.
Working on the same problem. So far I have shown that the property is true for m = 1. My line of thought is then to show that the property is true for m+1. I am however struggling with a double integral.
I do not remember exactly how N_m is characterized. That makes it a bit hard to integrate. (I know that there is an explicit formula for the B-splines, but good luck integrating that.) I was thinking that there must be a better way to do the integration.
So we are considering the statement:
As ase mentioned the statement is true for . This establishes the basis for the induction.
Now we assume that is true for a fixed value of . Using the induction hypothesis we want to derive . We get:
From here I'm not sure. This is where Ase got stuck (I presume). Suggestions are appreciated greatly. Thanks.
Thanks for the reply. Not sure if I get the convultion-idea. I know the definition of convultion and that by definition:
Could you maybe outline your comment one more time.
Furthermore, I was wondering if the intended could be shown via induction? Is that possible?
Now, just remember that for we define where the middle equality is by the change of variable theorem. With this definition in hand, and letting be as I defined it in the previous post we get:
Now, you're asked to prove that , but just note:
Where we use Fubini (the fact that the theorem applies is left to you) for the second equality and our induction hypothesis for the last.