📂Fourier Analysis

Explicit Formulas of B-splines

Formulas¹

For the function $f: \mathbb{R}\to \mathbb{R}$, let’s say

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

That is, $f_{+}$ is a function that replaces all parts of $f$ where the function value is less than $0$, with $0$. Also, let’s define

$$ f(x)_{+}^{n}:=\left( f(x)_{+} \right)^{n} $$

Then, for each $m=2,3,\dots$, the B-spline $N_{m}$ can be expressed as follows.

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

Proof

■