Los elementos de mayor grado pueden obtenerse mediante polinomios de Hermite $$ \begin{align*} H_{0i} &= [1 - 2 \ \ell_{(\xi_{i})}^{\prime} (\xi - \xi_{i})] [\ell_{(\xi)}]^{2} \\ H_{1i} &= (\xi - \xi_{i}) [\ell_{(\xi)}]^{2} \end{align*} $$
Usando la fórmula para polinomios de Lagrange $$ \begin{align*} \ell_{1} &= \frac{\xi - 1}{-1 - 1} = \frac{1}{2} - \frac{1}{2} \xi \\ \ell_{1}^{\prime} &= - \frac{1}{2} \\ \ell_{2} &= \frac{\xi - (-1)}{1 - (-1)} = \frac{1}{2} + \frac{1}{2} \xi \\ \ell_{2}^{\prime} &= \frac{1}{2} \end{align*} $$
Usando la fórmula para polinomios de Hermite: $$ \begin{align*} H_{01} &= \bigg\{ 1 - 2 \bigg[ -\frac{1}{2} \bigg] [\xi - (-1)] \bigg\} \bigg( \frac{1}{2} - \frac{1}{2} \xi \bigg)^{2} = \frac{1}{4} (\xi + 2) (\xi - 1)^{2} \\ H_{11} &= [\xi - (-1)] \bigg( \frac{1}{2} - \frac{1}{2} \xi \bigg)^{2} = \frac{1}{4} (\xi + 1) (\xi - 1)^{2} \\ H_{02} &= \bigg[ 1 - 2 \bigg( \frac{1}{2} \bigg) (\xi - 1) \bigg] \bigg( \frac{1}{2} + \frac{1}{2} \xi \bigg)^{2} = -\frac{1}{4} (\xi - 2) (\xi + 1)^{2} \\ H_{12} &= (\xi - 1) \bigg( \frac{1}{2} + \frac{1}{2} \xi \bigg)^{2} = \frac{1}{4} (\xi - 1) (\xi + 1)^{2} \end{align*} $$
Usando la fórmula para polinomios de Lagrange $$ \begin{align*} \ell_{1} &= \frac{\xi - 0}{-1 - 0} \frac{\xi - 1}{-1 - 1} = -\frac{1}{2} \xi + \frac{1}{2} \xi^{2} \\ \ell_{1}^{\prime} &= -\frac{1}{2} + \xi \\ \ell_{2} &= \frac{\xi - (-1)}{0 - (-1)} \frac{\xi - 1}{0 - 1} = 1 - \xi^{2} \\ \ell_{2}^{\prime} &= - 2 \xi \\ \ell_{3} &= \frac{\xi - 0}{1 - 0} \frac{\xi - 1}{1 - 1} = \frac{1}{2} \xi + \frac{1}{2} \xi^{2} \\ \ell_{3}^{\prime} &= \frac{1}{2} + \xi \end{align*} $$
Usando la fórmula para polinomios de Hermite $$ \begin{align*} H_{01} &= \bigg\{ 1 - 2 \bigg[ -\frac{1}{2} + (-1) \bigg] [\xi - (-1)] \bigg\} \bigg( -\frac{1}{2} \xi + \frac{1}{2} \xi^{2} \bigg)^{2} = \frac{1}{4} \xi^{2} (3 \xi + 4) (\xi - 1)^{2} \\ H_{11} &= [\xi - (-1)] \bigg( -\frac{1}{2} \xi + \frac{1}{2} \xi^{2} \bigg)^{2} = \frac{1}{4} \xi^{2} (\xi + 1) (\xi - 1)^{2} \\ H_{02} &= \{ 1 - 2 [ - 2 (0) ] (\xi - 0) \} \bigg( 1 - \xi^{2} \bigg)^{2} = (\xi - 1)^{2} (\xi + 1)^{2} \\ H_{12} &= (\xi - 0) \bigg( 1 - \xi^{2} \bigg)^{2} = \xi (\xi - 1)^{2} (\xi + 1)^{2} \\ H_{03} &= \bigg\{ 1 - 2 \bigg[ \frac{1}{2} + (1) \bigg] \bigg\} \bigg( \frac{1}{2} \xi + \frac{1}{2} \xi^{2} \bigg)^{2} = -\frac{1}{4} \xi^{2} (3 \xi - 4) (\xi + 1)^{2} \\ H_{13} &= [\xi - (1)] \bigg( \frac{1}{2} \xi + \frac{1}{2} \xi^{2} \bigg)^{2} = \frac{1}{4} \xi^{2} (\xi - 1) (\xi + 1)^{2} \end{align*} $$