best approximation polynomial $p_1(x)\in P_1$ for $x^3$

Question

I want to find a best approximation polynomial $p_1(x)\in P_1$ for $f(x)=x^3$ in $[-1,1]$ w.r.t. $||\cdot||_{\infty}$. I want to use Chebyshev polynomial to do that, but I don't know how to hang on.

Answer 1

Let $p(x)$ will be polynomial which you find. First note that $f(x)-p(x)$ is polynomial of degree $3$ with leading coefficient $1$. You know that among the polynomials of degree $3$ with leading coefficient $1$

$$w(x)=\frac{1}{4}T_{3}(x)$$

is the one of which the maximal absolute value on the interval $[−1, 1]$ is minimal, see.

So you know that you must have $f(x)-p(x)=\frac{1}{4}T_3(x)$, so $p(x)=x^3-\frac{1}{4}T_3(x)$.