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By Stone-Weierstrass theorem, the set of polynomial is dense in $C[a,b]$, I am wondering what is the sequence of polynomial which can approximate absolute value function $|x|$? I know using $\sqrt{x^{2}+1/n}$ can approximate it but it is not a polynomial.

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  • $\begingroup$ You can probably just pick $n$ points that range over the interval and construct the interpolating polynomial through those points. It won't be a nice sequence as $n$ increases, but I think it converges uniformly if you pick the points nicely. $\endgroup$ Jun 1, 2021 at 13:23
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    $\begingroup$ Weierstrass' can be proven constructively using the Bernstein polynomials. You can scale your problem to the interval $[0,1]$ and then take the combination in the link of the first few Bernstein polynomials. $\endgroup$
    – plop
    Jun 1, 2021 at 13:58

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For example, you can consider the binomial expansion of $\sqrt{1-y}$ on $[0,1]$. Namely, setting $y=1-x^2$, we have $$ |x|=\sqrt{1-y}=\sum_{m=0}^{\infty}\binom{1/2}{m}(-y)^m, \quad x\in[-1,1]. $$

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