Spline, B-Spline in Analysis
📂Fourier Analysis Spline, B-Spline in Analysis Definition If the function f : R → R f:\mathbb{R} \to \mathbb{R} f : R → R piecewise polynomial on interval R \mathbb{R} R spline on R \mathbb{R} R knots .
Explanation As can be seen from the definition, a spline does not have to be a continuous function . The following function f f f
f ( x ) = { 0 x ∈ [ ∞ , 0 ] 2 x 2 x ∈ ( 0 , 1 ] 2 − x x ∈ ( 1 , 4 ] 1 16 x 3 x ∈ ( 4 , ∞ ]
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}
f ( x ) = ⎩ ⎨ ⎧ 0 2 x 2 2 − x 16 1 x 3 x ∈ [ ∞ , 0 ] x ∈ ( 0 , 1 ] x ∈ ( 1 , 4 ] x ∈ ( 4 , ∞ ]
In the above case, x = 0 x=0 x = 0 x = 1 x=1 x = 1 x = 4 x=4 x = 4 B-spline is a spline with good properties. A B B B N 1 N_{1} N 1 indicator function on interval [ 0 , 1 ] [0,1] [ 0 , 1 ]
N 1 ( x ) : = χ [ 0 , 1 ] ( x ) , x ∈ R
N_{1}(x) :=\chi_{[0,1]}(x)\quad , x\in \mathbb{R}
N 1 ( x ) := χ [ 0 , 1 ] ( x ) , x ∈ R
And for m ∈ N m \in \mathbb{N} m ∈ N N m + 1 N_{m+1} N m + 1
N m + 1 ( x ) : = ( N m ∗ N 1 ) ( x )
\begin{equation} N_{m+1}(x) := (N_{m} * N_{1})(x)\end{equation}
N m + 1 ( x ) := ( N m ∗ N 1 ) ( x )
Here, ∗ \ast ∗ convolution . m m m order of the B-spline N m N_{m} N m ( 1 ) (1) ( 1 )
N m = N m − 1 ∗ N 1 = N m − 2 ∗ N 1 ∗ N 1 = N m − 3 ∗ N 1 ∗ N 1 ∗ N 1 = N 1 ∗ N 1 ∗ N 1 ⋯ ∗ N 1 ⏟ m
\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*}
N m = N m − 1 ∗ N 1 = N m − 2 ∗ N 1 ∗ N 1 = N m − 3 ∗ N 1 ∗ N 1 ∗ N 1 = m N 1 ∗ N 1 ∗ N 1 ⋯ ∗ N 1
Also, from the definition of N 1 N_{1} N 1 convolution , the following formula holds true:
N m + 1 ( x ) = ∫ − ∞ ∞ N m ( x − t ) N 1 ( t ) d t = ∫ 0 1 N m ( x − t ) d t
N_{m+1}(x)=\int _{-\infty} ^{\infty}N_{m}(x-t)N_{1}(t)dt=\int_{0}^{1}N_{m}(x-t)dt
N m + 1 ( x ) = ∫ − ∞ ∞ N m ( x − t ) N 1 ( t ) d t = ∫ 0 1 N m ( x − t ) d t
The picture below shows the graphs of N 2 N_{2} N 2 N 3 N_{3} N 3
A B-spline of order m ∈ N m\in \mathbb{N} m ∈ N
(a) s u p p N m = [ 0 , m ] \mathrm{supp}N_{m}=[0,m] supp N m = [ 0 , m ] and \text{and} and N m ( x ) > 0 for x ∈ ( 0 , m ) N_{m}(x)>0 \text{ for } x\in(0,m) N m ( x ) > 0 for x ∈ ( 0 , m )
(b) ∫ − ∞ ∞ N m ( x ) d x = 1 \displaystyle \int _{-\infty} ^{\infty} N_{m}(x)dx=1 ∫ − ∞ ∞ N m ( x ) d x = 1
(c) For m ≥ 2 m\ge 2 m ≥ 2
∑ k ∈ Z N m ( x − k ) = 1 , ∀ x ∈ R
\begin{equation} \sum \limits_{k \in \mathbb{Z}} N_{m}(x-k)=1,\quad \forall x\in \mathbb{R} \end{equation}
k ∈ Z ∑ N m ( x − k ) = 1 , ∀ x ∈ R
(c’) When m = 1 m=1 m = 1 x ∈ R ∖ Z x\in \mathbb{R}\setminus \mathbb{Z} x ∈ R ∖ Z
See Also 
