What is a polynomial approximation?

Yves Daoust provided an approximation using a Taylor series, which gives a good approximation near one point.

Here is an illustration of Chebyshev's equioscillation theorem: it's possible to find a polynomial that is a good approximation on a whole interval.

For instance, an approximation of degree $3$ of $sin(x)$ on $[-pi,pi]$ is given by the polynomial:

$$- 0.0000000534717177+ 0.8245351702,x+ 0.00000008122122864,{x}^{2}\- 0.08691931647,{x}^{3}$$

(this approximation was found using Maple)

Here is a plot of this polynomial and $sin(x)$:

These minimax approximations have the nice property that the difference with the function oscillates between two bounds. Here is the difference:

Suppose you have a function $f$ defined on a certain domain. A polynomial approximation of $f$ is a polynomial $p$ that is the most closest approximation to $f$ given certain conditions..

Now, of course if $f$ is not a polynomial and the domain is $mathbb{R}$ then $p$ will never get close enough to $f$. But for example if you consider a compact interval and a function $f$ that is quite regular, then the magic happens: this is Weierstrass' theorem.

Note: What I wrote before is quite vague, so let be more clear: Consider the interval $[a,b] subset mathbb R$ and a function $f: [a,b] to mathbb R$. We say that a polynomial $p$ is a good approximation of $f$ if the maximum distance from $p$ to $f$ all over $[a,b]$ is less than a tolerance we decide, or in math language $$ forall x in [a,b] , , ,text{we have} , , , |p(x) - f(x) | < TOL $$

Weierstrass' theorem says: Given a continous function $f:[a,b] to mathbb R$ then there exists a sequence of polynomials $(p_n)$ that converges uniformly to $f$ on the interval $[a,b]$. In other terms: $$lim_{n to infty} sup_{xin [a,b]} |p(x) - f(x) | = 0$$

This is only an example. There exists so many theorems about approximation.