Exponential Hermite basis#
See [Uhlmann2014].
(1)#\[\begin{split}\phi_{1,M}(t) &= \left\{
\begin{array}{ll}
g_{1,M}(t) & t \geq 0 \\
g_{1,M}(-t) & t < 0
\end{array}
\right.\end{split}\]
(2)#\[\begin{split}\phi_{2,M}(t) &= \left\{
\begin{array}{lll}
g_{2,M}(t) & t \geq 0 \\
-g_{2,M}(-t) & t < 0
\end{array}
\right.\end{split}\]
where \(M\) is the number of control points and
(3)#\[\begin{split}g_{1,M}(t)=
\left\{
\begin{array}{ll}
a_1(M) + b_1(M) t + c_1(M) \text{e}^{\text{j}\,\frac{2\,\pi}{M}\,t} + d_1(M) \text{e}^{-\text{j}\,\frac{2\,\pi}{M}\,t} & t \in [0,1[ \\
0 & \mbox{elsewhere}
\end{array}
\right.\end{split}\]
(4)#\[\begin{split}g_{2,M}(t)=
\left\{
\begin{array}{ll}
a_2(M) + b_2(M) t + c_2(M) \text{e}^{\text{j}\,\frac{2\,\pi}{M} t} + d_2(M) \text{e}^{-\text{j}\,\frac{2\,\pi}{M} t} & t \in [0,1[ \\
0 & \mbox{elsewhere} \end{array}
\right.\end{split}\]
(5)#\[\begin{split}\begin{array}{ll}
a_1(M) = \frac{\text{j}\,\frac{2\,\pi}{M}+1 + \text{e}^{\text{j}\,\frac{2\,\pi}{M}} (\text{j}\,\frac{2\,\pi}{M}-1)}{q(M)} & b_1(M) = -\frac{\text{j}\,\frac{2\,\pi}{M} (\text{e}^{\text{j}\,\frac{2\,\pi}{M}} + 1)}{q(M)} \\
c_1(M) = \frac{1}{q(M)} & d_1(M) = -\frac{\text{e}^{\text{j}\,\frac{2\,\pi}{M}}}{q(M)} \\
a_2(M)= \frac{p(M)}{\text{j}\,\frac{2\,\pi}{M} (\text{e}^{\text{j}\,\frac{2\,\pi}{M}}-1) q(M) } & b_2(M)= -\frac{\text{e}^{\text{j}\,\frac{2\,\pi}{M}}-1}{q(M)} \\
c_2(M)= \frac{\text{e}^{\text{j}\,\frac{2\,\pi}{M}}-\text{j}\,\frac{2\,\pi}{M}-1}{\text{j}\,\frac{2\,\pi}{M}(\text{e}^{\text{j}\,\frac{2\,\pi}{M}}-1)q(M)} &
d_2(M)= -\frac{\text{e}^{\text{j}\,\frac{2\,\pi}{M}} (\text{e}^{\text{j}\,\frac{2\,\pi}{M}}(\text{j}\,\frac{2\,\pi}{M}-1) + 1)}{\text{j}\,\frac{2\,\pi}{M}(\text{e}^{\text{j}\,\frac{2\,\pi}{M}} - 1) q(M)}
\end{array}\end{split}\]
(6)#\[\begin{split}p(M) &= \text{j}\,\frac{2\,\pi}{M}+1+\text{e}^{\text{j}\,\frac{4\,\pi}{M}}(\text{j}\,\frac{2\,\pi}{M}-1)\, \\
q(M) &= \text{j}\,\frac{2\,\pi}{M}+2+\text{e}^{\text{j}\,\frac{2\,\pi}{M}}(\text{j}\,\frac{2\,\pi}{M}-2)\end{split}\]
