Polynomial basis (B-spline)#

The polynomial spline basis of order $n$, usually denoted as $\beta_n$, is obtained by convolving the $0$-th order basis

(1)#\[\begin{split}\beta_0(t)=\begin{cases} 1, \quad t\in [-\frac{1}{2}, \frac{1}{2}] \\ 0, \quad \mathrm{elsewhere} \end{cases}\end{split}\]

with itself $n$ times, emph{i.e.}

(2)#\[\beta_n(t) = (\smash[b]{\underbrace{\beta_0 \ast ... \ast \beta_0}_\text{$n$ times}}).\]

See~cite{unser1999} for more details.

Linear ($1^{\text{st}}$ order) polynomial basis

(3)#\[\begin{split}\beta_1(t)=\begin{cases} 1-|t|, \quad |t|\in [0, 1] \\ 0, \quad \mathrm{elsewhere} \end{cases}\end{split}\]

Quadratic ($2^{\text{nd}}$ order) polynomial basis

(4)#\[\begin{split}\beta_2(t)=\begin{cases} \frac{1}{2}t^2 + \frac{3}{2}t + \frac{9}{8}, \quad t\in [-\frac{3}{2}, -\frac{1}{2}[ \\ \frac{3}{4}-t, \quad t\in [-\frac{1}{2}, \frac{1}{2}[ \\ \frac{1}{2}t^2 - \frac{3}{2}t + \frac{9}{8}, \quad t\in [\frac{1}{2}, \frac{3}{2}] \\ 0, \quad \mathrm{elsewhere} \end{cases}\end{split}\]

Cubic ($3^{\text{rd}}$ order) polynomial basis

(5)#\[\begin{split}\beta_3(t)=\begin{cases} \frac{2}{3} - |t|^2 + \frac{1}{2}|t|^3, \quad |t|\in [0, 1[ \\ \frac{1}{6}(2 - |t|)^3, \quad |t| \in [1, 2] \\ 0, \quad \mathrm{elsewhere} \end{cases}\end{split}\]