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I'm trying to show that any continuous function $f$ with period $2\pi$ can be approximated by a fourier series $P$ (i.e. $| P(x) - f(x)| < \epsilon $ for $\epsilon > 0$). Any suggestions?

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The inequality you wrote down for convergence seems to imply that you are interested in uniform convergence. In general the Fourier series of a continuous function won't converge uniformly. It is true that trigonometric polynomials are dense in $C^0$ with respect to uniform convergence, but this does not imply that the Fourier series converges uniformly. After a rearrangement of terms it will, however, converge uniformly. See the link provided by Boston. – user20266 Mar 17 '12 at 18:24
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If it is uniform approximation that you want, then you can use Fejer's Theorem.

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Here's a nice section of convergence of fourier series for piece-wise continuous functions: Have a read through, and let us know if you need more help!

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