Spline, B-Spline in Analysis 📂Fourier Analysis

Spline, B-Spline in Analysis

Definition¹

If the function $f:\mathbb{R} \to \mathbb{R}$ is a piecewise polynomial on interval $\mathbb{R}$, it is called a spline on $\mathbb{R}$. The points where the polynomial changes are called knots.

Explanation

As can be seen from the definition, a spline does not have to be a continuous function. The following function $f$ is an example of a spline.

$$ f(x) = \begin{cases} 0 & x\in[\infty,0] \\ 2x^{2}&x\in(0,1] \\ 2-x & x\in (1,4] \\ \frac{1}{16}x^{3} & x\in(4,\infty] \end{cases} $$

In the above case, $x=0$, $x=1$, and $x=4$ are knots. A B-spline is a spline with good properties. A $B$-B-spline $N_{1}$ is defined using the indicator function on interval $[0,1]$ as follows:

$$ N_{1}(x) :=\chi_{[0,1]}(x)\quad , x\in \mathbb{R} $$

And for $m \in \mathbb{N}$, B-spline $N_{m+1}$ is defined as follows:

$$ \begin{equation} N_{m+1}(x) := (N_{m} * N_{1})(x)\end{equation} $$

Here, $\ast$ is a convolution. $m$ is called the order of the B-spline $N_{m}$. By definition $(1)$, the following is true:

$$ \begin{align*} N_{m} &= N_{m-1} \ast N_{1} \\ &= N_{m-2} \ast N_{1} \ast N_{1} \\ &= N_{m-3} \ast N_{1} \ast N_{1}\ast N_{1} \\ &= \underbrace{N_{1} \ast N_{1} \ast N_{1} \cdots \ast N_{1}}_{m} \end{align*} $$

Also, from the definition of $N_{1}$ and convolution, the following formula holds true:

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

The picture below shows the graphs of $N_{2}$ and $N_{3}$ from left to right.

Properties

A B-spline of order $m\in \mathbb{N}$ satisfies the following properties:

(a) $\mathrm{supp}N_{m}=[0,m]$ $\text{and}$ $N_{m}(x)>0 \text{ for } x\in(0,m) $

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

(c) For $m\ge 2$, the following formula holds true:

$$ \begin{equation} \sum \limits_{k \in \mathbb{Z}} N_{m}(x-k)=1,\quad \forall x\in \mathbb{R} \end{equation} $$

(c’) When $m=1$, the above formula holds true for $x\in \mathbb{R}\setminus \mathbb{Z}$.