Exponential

class splinebox.basis_functions.Exponential(M)

???

\[\begin{split}f(t) = \begin{cases} \frac{L}{\alpha^2} 2 \sin(\frac{\alpha}{2} t)^2 & \text{for } 0 \leq t < 1 \\ \frac{L}{\alpha^2} (\cos(\alpha (t - 2)) + \cos(\alpha (t - 1)) - 2 \cos(\alpha)) & \text{for } 1 \leq t < 2 \\ \frac{L}{\alpha^2} 2 \sin(\frac{\alpha}{2} (t - 3))^2 & \text{for } 2 \leq t \leq 3 \\ 0 & \text{otherwise}\end{cases}\end{split}\]

\[\text{where } L=(\frac{\sin(\pi / M)}{\pi / M})^{-2}\text{ and }t = t + \frac{support}{2}\]

\[\begin{split}f(t) = \begin{cases} 2 L \sin(\alpha t)^2 & \text{for } 0 \leq t < 1 \\ L (\cos(2\alpha (t - 2)) + \cos(2\alpha (t - 1)) - 2 \cos(2\alpha)) & \text{for } 1 \leq t < 2 \\ 2 L \sin(\alpha (t - 3))^2 & \text{for } 2 \leq t \leq 3 \\ 0 & \text{otherwise}\end{cases}\end{split}\]

\[\text{where } \alpha=\frac{\pi}{M}, L=\frac{1}{4 \sin(\alpha)^2} \text{ and }t = t + \frac{3}{2}\]