A note on Taylor expansion of real function

Abstract

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 $0\le k\le n$, 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 $\le k$, we show that $S_{k,x_0}(x)$ is the limit of the best approximations of $f(x)$ on $[x_0-\epsilon ,x_0+\epsilon ]$ as $\epsilon \to 0$.

8 pages, title was changed to a more specific one