Legendre多项式
\(\mathfrak{R}_{2}([-1,1],\mathbb{R})\) 上的多项式函数系的基 \(\{1,x,x^{2},\cdots,x^{n},\cdots\}\) 线性无关但并非规范正交基
对其作规范正交化:
\[ p_{1}=1,\qquad p_{2}=\frac{\sqrt{6}}{2}x,\qquad p_{n}=\frac{x^{n}-\sum\limits_{k=1}^{n-1}\int_{-1}^{1}x^{n+k}\,\mathrm{d}x}{||x^{n}-\sum\limits_{k=1}^{n-1}\int_{-1}^{1}x^{n+k}\,\mathrm{d}x||} \]
计算 \(p_{n}\) ,
\[\begin{align}p_{n}&=\frac{x^{n}-\sum\limits_{k=1}^{n-1}\frac{1-(-1)^{n+k+1}}{n+k+1}x^{k}}{||x^{n}-\sum\limits_{k=1}^{n-1}\frac{1-(-1)^{n+k+1}}{n+k+1}x^{k}||}\end{align} \]