📝 题目
5.5.5 设 $f\left( x\right) \in R\left\lbrack {-\pi ,\pi }\right\rbrack$ ,求函数
$$ F\left( {{\alpha }_{0},{\alpha }_{1},\cdots ,{\alpha }_{n},{\beta }_{1},\cdots ,{\beta }_{n}}\right) $$
$$ = \frac{1}{\pi }{\int }_{-\pi }^{\pi }{\left\lbrack f\left( x\right) - \frac{{\alpha }_{0}}{2} - \mathop{\sum }\limits_{{k = 1}}^{n}\left( {\alpha }_{k}\cos kx + {\beta }_{k}\sin kx\right) \right\rbrack }^{2}\mathrm{\;d}x $$
的最小值.
💡 答案与解析
5.5.5 $\left\{ \begin{array}{ll} {\alpha }_{k} = \frac{1}{\pi }{\int }_{-\pi }^{\pi }f\left( x\right) \cos {kx}\mathrm{\;d}x & \left( {k = 0,1,\cdots ,n}\right) , \\ {\beta }_{k} = \frac{1}{\pi }{\int }_{-\pi }^{\pi }f\left( x\right) \sin {kx}\mathrm{\;d}x & \left( {k = 1,2,\cdots ,n}\right) . \end{array}\right.$