B-Spline Definition

I'm currently working on my master's project. For this, I rely on one PhD-thesis in which I found a statement I do not understand. Unfortunately, the author hasn't answered to my mails yet, so I have to ask you. It's about the definition of B-Splines.

As a quicknote: We are looking to create a symmetric density function with zero mean which is smooth, has finite support and the integral over it is one.

The author proposes quadratic B-splines with the following statement:

A B-Spline $B_i$ of unit width is given by

$B_0(x)=\left\{\begin{array}{ll} 1, & for - \frac 1 2 \leq x \leq \frac 1 2 \\ 0, & otherwise \end{array}\right. .$

$B_{i+1}(x) = 2 B_{\lfloor i/2 \rfloor}(2x) \ast 2 B_{\lceil i/2 \rceil} (2x)$

For example, the resulting B-spline is given by:

$\rho(x)=\left\{\begin{array}{ll} \frac{-27 x^2 + 36}{16}, & 0 \leq x < \frac 1 6 \\ \frac{27 x^2 - 108x + 108}{32}, & \frac 1 6 \leq x < 1/2 \\ 0, & otherwise\end{array}\right. .$

Ok, this is was given in the thesis. Now there are a couple of things I do not understand: First of all, the definition: I have never seen this definition before and couldn't find it anywhere. I know about cardinal B-Splines, but they are defined via a convolution with $B_0$. So my question is: Could anyone explain me, what was done here?

Second question is the explicit formula: I am looking for an explicit formula, since I am implementing this in a simulation. However, this seems just wrong, as neither is the integral 1, nor is it even continuous (jump at 1/6). So can maybe anyone explain where this comes from?

• I've never seem a b-spline constructed that way. The formula for $\rho$ seems to have several problems. It contains a mysterious"$r$" term. It's obviously not continuous at either $x=0$ or $x=0.5$. – bubba Feb 28 '13 at 3:45
• Are you in love with b-splines? A piece of a trig function might be easier to deal with. Like $\sin(x)$ for $-\pi/2 \le x \le 3\pi/2$. Obviously you'd have to shift it and scale it, but that's easy enough. – bubba Feb 28 '13 at 3:48
• Sorry, that "r" was a mistake of mine, it is actually justa n x. – disaster Feb 28 '13 at 11:25