[Math] Using Stone-Weierstrass Theorem to show that trig polynomials are dense in $L^2([0,2\pi])$

fourier analysisreal-analysis

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!

Best Answer

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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