论文标题
关于泰勒扩展实际功能的注释
A note on Taylor expansion of real function
论文作者
论文摘要
令$ f（x）$为一个真实函数，在间隔$ [a，b] $上具有$（n+1）$ - th衍生物。对于任何点$ x_0 \ in（a，b）$和任何整数$ 0 \ leq k \ leq n $，用$ s_ {k，x_0}（x）$ k $ the $ k $ - taylor扩展的$ f（x）$ at $ x_0 $的$ f（x）$。 $ s_ {k，x_0}（x）= \ sum_ {i = 0}^k \ frac {f^{（i）}（i）}（x_0）} {i！}（x-x_0）^i。$ $ $ $ $在本注中，我们考虑$ l_2 $ approximation of $ l_2 $ f（x） $ s_ {k，x_0}（x）$是$ [x_0- \ varepsilon，x_0+\ varepsilon] $ as $ \ varepsilon \ to $ \ varepsilon \至0 $的最佳近似值的极限。
Let $f(x)$ be a real function which has $(n+1)$-th derivative on an interval $[a, b]$. For any point $x_0\in (a, b)$ and any integer $0\leq k\leq n$, denote by $S_{k,x_0}(x)$ the $k$-th truncation of the Taylor expansion of $f(x)$ at $x_0$, i.e. $$S_{k,x_0}(x)=\sum_{i=0}^k\frac{f^{(i)}(x_0)}{i!}(x-x_0)^i.$$ In this note, we consider the $L_2$-approximation of $f(x)$ by polynomials of degree $\leq k$, we show that $S_{k,x_0}(x)$ is the limit of the best approximations of $f(x)$ on $[x_0-\varepsilon, x_0+\varepsilon]$ as $\varepsilon\to 0$.
