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Suppose I want to construct a sequence of polynomials that converges to $|x|$ pointwise.

I am pretty good on proving that sequences of functions converge to things pointwise, but I am having trouble actually coming up with a sequence of polynomials that converges to $|x|$. Someone told me to try using the Taylor series expansion of $(1−x)^{1/2}$, but that just confused me, since I am very rusty on Taylor series. Would anyone mind explaining this to me a little?

Thanks!!

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You can use Chebyshev polynomials of the first kind.

$$ P_{2n}(x)=\frac{2}{\pi}-\sum_{j=1}^{n}\frac{4(-1)^j}{\pi(4j^2-1)}T_{2j}(x)$$

converges uniformly to $|x|$ over the compact set $[-1,1]$, in virtue of the Fourier-Chebyshev series (the Chebyshev polynomials of the first kind are a complete base of orthonormal functions wrt the scalar product $<f,g>=\int_{-1}^{1}\frac{f(x)g(x)dx}{\sqrt{1-x^2}}$).

This choice is quite good since:

$$ \sup_{x\in(-1,1)}||x|-P_{2n}(x)|=O\left(\frac{1}{n}\right).$$

See also: Weierstrass approximation Theorem.

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