reemplazando los valores \( \xi_{1} = -1 \) y \( \xi_{2} = 1 \) $$ \begin{align*} \alpha_{0} + \alpha_{1} (-1) + \alpha_{2} (-1)^{2} + \alpha_{3} (-1)^{3} &= v_{1} \\ \alpha_{1} + 2 \alpha_{2} (-1) + 3 \alpha_{3} (-1)^{2} &= \theta_{1} \\ \alpha_{0} + \alpha_{1} (1) + \alpha_{2} (1)^{2} + \alpha_{3} (1)^{3} &= v_{2} \\ \alpha_{1} + 2 \alpha_{2} (1) + 3 \alpha_{3} (1)^{2} &= \theta_{2} \end{align*} $$
simplificando $$ \begin{align*} \alpha_{0} - \alpha_{1} + \alpha_{2} - \alpha_{3} &= v_{1} \\ \alpha_{1} - 2 \alpha_{2} + 3 \alpha_{3} &= \theta_{1} \\ \alpha_{0} + \alpha_{1} + \alpha_{2} + \alpha_{3} &= v_{2} \\ \alpha_{1} + 2 \alpha_{2} + 3 \alpha_{3} &= \theta_{2} \end{align*} $$
en forma matricial $$ \begin{equation*} \begin{bmatrix} 1 & -1 & 1 & -1 \\ 0 & 1 & -2 & 3 \\ 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} = \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \end{equation*} $$
resolviendo $$ \begin{equation*} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} = \begin{bmatrix} \frac{1}{2} & \frac{1}{4} & \frac{1}{2} & -\frac{1}{4} \\ -\frac{3}{4} & -\frac{1}{4} & \frac{3}{4} & -\frac{1}{4} \\ 0 & -\frac{1}{4} & 0 & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & -\frac{1}{4} & \frac{1}{4} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \end{equation*} $$
reemplazando $$ \begin{align*} v &= \begin{bmatrix} 1 & \xi & \xi^{2} & \xi^{3} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} \\ &= \begin{bmatrix} 1 & \xi & \xi^{2} & \xi^{3} \end{bmatrix} \begin{bmatrix} \frac{1}{2} & \frac{1}{4} & \frac{1}{2} & -\frac{1}{4} \\ -\frac{3}{4} & -\frac{1}{4} & \frac{3}{4} & -\frac{1}{4} \\ 0 & -\frac{1}{4} & 0 & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} & -\frac{1}{4} & \frac{1}{4} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \\ &= \begin{bmatrix} \frac{1}{4} (\xi + 2) (\xi - 1)^{2} & \frac{1}{4} (\xi + 1) (\xi - 1)^{2} & -\frac{1}{4} (\xi - 2) (\xi + 1)^{2} & \frac{1}{4} (\xi - 1) (\xi + 1)^{2} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \end{align*} $$
Reescribiendo \( v \): $$ \begin{align*} v &= \bigg[ \frac{1}{4} (\xi + 2) (\xi - 1)^{2} \bigg] v_{1} + \bigg[ \frac{1}{4} (\xi + 1) (\xi - 1)^{2} \bigg] \theta_{1} + \bigg[ -\frac{1}{4} (\xi - 2) (\xi + 1)^{2} \bigg] v_{2} + \bigg[ \frac{1}{4} (\xi - 1) (\xi + 1)^{2} \bigg] \theta_{2} \\ &= N_{1} v_{1} + N_{2} \theta_{1} + N_{3} v_{2} + N_{4} \theta_{2} \end{align*} $$