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Are there sequences of (real) polynomials ($p_n$) that converge on the absolute value:

$\lim\limits_{n\rightarrow\infty} p_n(x) = |x|$?

If so, what is it/are they? if not, why not?

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2 Answers 2

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Yes, if by "converge" you mean in the pointwise sense. In fact, any continuous function $f : \mathbb{R} \to \mathbb{R}$ may be realized as a pointwise limit of a sequence of polynomials. To see this, you can use the Weierstrass approximation theorem to take $p_n$ to be a polynomial on $\mathbb{R}$ such that $\max\limits_{-n \leq x \leq n} |p_n(x) - f(x)| < \frac{1}{n}$.

As for what the polynomials themselves are, there is a constructive proof of the Weierstrass theorem using Bernstein polynomials that you might be able to use to get a handle on them:

https://en.wikipedia.org/wiki/Bernstein_polynomial

This is just one idea. I would imagine that something even simpler would work in the case $f(x) = |x|$ (but I could be wrong).

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THE STONE-WEIERSTRASS THEOREM

If $f$ is a continuous complex function on $[a, b]$, there exists a sequence of polynolnials $P_{n}$ such that

$lim P_{n}(x) = f(x)$

uniformly on $[a, b]$. If $f$ is real, the $P_{n}$ may be taken real.

see "Principals_of_mathematical_analysis-Walter_Rudin" p.159 and p.161

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