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I am trying to show use Stone-Weierstrass Theorem to show that trig polynomials are dense in $L^2([0,2\pi])$, however the following things concerns me.

  1. trig polynomails doesn't separate points since $f(0) = f(2\pi)$. My solution is to show that trig polynomial is dense in all $[0,2\pi - \epsilon]$, then let $\epsilon \rightarrow 0$, we could say that trig polynomials are dense almost everywhere in $[0,2\pi]$. (I doubt if this is correct)

  2. Stone-Weierstrass theorem only provide density in $C(K)$ (all complex continuous functions on compact K). How could we extend this to the space of $L^2$?

I hope you could give me hints on these. Thank you!

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  • $\begingroup$ As far as I know a real trigonometric polynomial is a function of the form: $f(x) = a_0 + \Sigma_{j=1}^{n} a_n \cos(jx) + \Sigma_{j=1}^{n} b_n \cos(jx) $. This clearly shows that as an algebra it separates points. $\endgroup$ Feb 19, 2019 at 6:14
  • $\begingroup$ As an aside note, not only they are dense but we have a formula for constructing this trig polynomials that approximate (a continuous function in this case) which is quite important. As Kavi Rama Murthy said, in $L^2$ the continuous functions are dense. See this post: math.stackexchange.com/questions/226049/… $\endgroup$ Feb 19, 2019 at 6:21

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Apply Stone - Weierstrass Theorem to $C(T)$ where $T$ is the unit circle. Trig. polynomials do separate points here.

You have to use two more facts:

a) unform convergence implies convergence in $L^{2}$ and

b) $L^{2}$ functions can be approximated by continuous functions.

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  • $\begingroup$ Thank you for the answer! I wonder if you could further explain what it means to apply it to $C(T)$? and also could you provide the reference for the second fact? Thank you! $\endgroup$ Feb 19, 2019 at 19:43
  • $\begingroup$ The circle $T$ is a compact Hausdorff space and a continuous periodic function $f$ on $[0,2\pi]$ corresponds to a continous function $g$ on $T$ by the formula $g(e^{it})=f(t)$. Part b) of my answer is very standard and a proof can be found in any book on real analysis. In particular Rudin's RCA has a proof. $\endgroup$ Feb 19, 2019 at 23:08

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