📂Fourier Analysis

Regularity of B-Splines

Theorem¹

For $m=2,3,\dots$, the B-spline $N_{m}$ has the following properties.

(a) $N_{m}\in C^{m-2}(\mathbb{R})$

(b) For $k\in \mathbb{Z}$, in each interval $[k,k+1]$, $N_{m}$ is at most a polynomial of degree $m-1$.

Explicit formula of B-spline

$$N_{m}(x) = \frac{1}{(m-1)!}\sum \limits_{j=0}^{m} \left( -1 \right)^{j}\binom{m}{j}\left( x-j \right)_{+}^{m-1},\quad x\in \mathbb{R}$$

Where

$$f(x)_{+}:=\max \left( 0,f(x) \right) \quad \& \quad f(x)_{+}^{n}:=\left( f(x)_{+} \right)^{n}$$

Lemma

For $m=2,3,\cdots$, $x_{+}^{m-1}$ is differentiable up to $m-2$ times, and the $m-2$th derivative is continuous.

Proof

For $m=2$,

$$x_{+}^{1}=\max(0,x)=\begin{cases} 0 & \text{if}\ x\le0 \\ x & \text{if}\ x\ge0 \end{cases}$$

so it is continuous at all points, and differentiable except at $x=0$. For $m=3$,

$$x_{+}^{2}=\left( \max(0,x) \right)^{2}=\begin{cases} 0 & \text{if}\ x\le0 \\ x^{2} & \text{if}\ x\ge0 \end{cases}$$

so it is differentiable at all points. The derivative is $2x_{+}^{1}$, so it is continuous at all points, and differentiable except at $x=0$.

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Proof

(a)

By the explicit formula of the B-spline, $N_{m}$ is a linear combination of translations of $x_{+}^{m-1}$. Therefore, according to the lemma below, $N_{m}$ is differentiable up to $m-2$ times, and each of its derivatives is continuous.

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(b)

For each $j=0,1,\dots,m$, the following formula holds.

$$(x-j)_{+}^{m-1}=\left( \max \left( 0,x-j \right) \right)^{m-1}=\begin{cases} 0 &\text{if}\ x\le j \\ (x-j)^{m-1}& \text{if}\ x>j \end{cases}$$

Since $N_{m}$ is a linear combination of such functions, it trivially holds.

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