1. Porperty of B-spline

Hello,

Assuming that a B-spline $\displaystyle N_{m}$ is defined for some $\displaystyle m \in \mathbb{N}$ then the B-spline $\displaystyle N_{m+1}$ is defined by:

$\displaystyle N_{m+1}(x)=\int_{0}^{1}N_{m}(x-t))dt$

How can I, through induction, show that:

$\displaystyle \int_{-\infty}^{\infty}N_{m}(x)dx=1$

Suggestions would be greatly appreciated.

2. How is $\displaystyle N_1(x)$ defined?

3. Hi

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.

4. Hi HallsofIvy

N_1(x) is the characteristic function from [0,1].

5. B-spline

Hello,

So we are considering the statement:

$\displaystyle S_{m}: \hspace{0,5cm}\int_{-\infty}^{\infty}N_{m}(x)dx=1$

As ase mentioned the statement $\displaystyle S_{1}$ is true for $\displaystyle m=1$. This establishes the basis for the induction.

Now we assume that $\displaystyle S_{m}$ is true for a fixed value of $\displaystyle m$. Using the induction hypothesis we want to derive $\displaystyle S_{m+1}$. We get:

$\displaystyle \int_{-\infty}^{\infty}N_{m+1}(x)dx=\int_{-\infty}^{\infty}\int_{0}^{1}N_{m}(x-t)dt$

From here I'm not sure. This is where Ase got stuck (I presume). Suggestions are appreciated greatly. Thanks.

6. Originally Posted by surjective
Hello,

Assuming that a B-spline $\displaystyle N_{m}$ is defined for some $\displaystyle m \in \mathbb{N}$ then the B-spline $\displaystyle N_{m+1}$ is defined by:

$\displaystyle N_{m+1}(x)=\int_{0}^{1}N_{m}(x-t))dt$

How can I, through induction, show that:

$\displaystyle \int_{-\infty}^{\infty}N_{m}(x)dx=1$

Suggestions would be greatly appreciated.
Are you familiar with the convolution of functions? because if so this problem is easy since $\displaystyle N_m = f\ast N_{m-1}$ where $\displaystyle f=1_{[0,1]}$ and it's well known that since both $\displaystyle f$ and $\displaystyle N_{m-1}$ are in $\displaystyle L^1(\mathbb{R} )$ then $\displaystyle \int_{\mathbb{R} } N_m = \left( \int_{\mathbb{R} } f \right) \left( \int_{\mathbb{R} } N_{m-1} \right) = 1$

7. Property revisited

Thanks for the reply. Not sure if I get the convultion-idea. I know the definition of convultion and that by definition:

$\displaystyle N_{m}=N_{1}*N_{1}*N_{1}*\cdots *N_{1}$

Could you maybe outline your comment one more time.

Furthermore, I was wondering if the intended could be shown via induction? Is that possible?

8. Originally Posted by surjective
Furthermore, I was wondering if the intended could be shown via induction? Is that possible?
Notice that the idea for a proof I gave uses an inductive argument.

Now, just remember that for $\displaystyle f,g\in L^1(\mathbb{R} )$ we define $\displaystyle f\ast g(x)= \int_{\mathbb{R} } f(x-t)g(t)dt = \int_{ \mathbb{R} } g(x-t)f(t)dt = g\ast f (x)$ where the middle equality is by the change of variable theorem. With this definition in hand, and letting $\displaystyle f$ be as I defined it in the previous post we get:

$\displaystyle f\ast N_{m-1}(x) = \int_{ \mathbb{R} } N_{m-1}(x-t)f(t)dt = \int_{0}^{1} N_{m-1}(x-t)dt = N_m(x)$.

Now, you're asked to prove that $\displaystyle \int_{ \mathbb{R} } f\ast N_{m-1} =1$ , but just note:

$\displaystyle \int_{\mathbb{R} } f\ast N_{m-1} = \int_ {\mathbb{R} } \left(\int_{\mathbb{R} } f(x-t)N_{m-1}(t)dt\right) dx$$\displaystyle = \int_{\mathbb{R} } \left( \int_{\mathbb{R} } f(x-t)N_{m-1}(t)dx\right) dt = \left( \int_{\mathbb{R} } f(y)dy \right) \left( \int_{\mathbb{R} } N_{m-1}(t)dt \right) =1$

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.

9. Property

Thanks for the response.

It seems that you are well versed in this subject. I have posted another questions elsewhere about the centered B-spline. Would it be possible for you to have a look at it?