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From page 53 of Fourier Analysis by Stein, we have

If $f$ is integrable on the circle, then the Fourier series of $f$ is Cesaro summable to $f$ at every point of continuity of $f$.

Moreover, if $f$ is continuous on the circle, then the Fourier series of $f$ is uniformly Cesaro summable to $f$.

Then it immediately has the following corollary which I don't understand:

Continuous functions on the circle can be uniformly approximated by trigonometric polynomials.

I mean, partial sum of Fourier series is a trigonometric. If you can approximate $f$ uniformly using partial sum of its Fourier series, then of course you can uniformly approximate by trigonometric polynomials. But then it feels like the corollary is not saying anything...Perhaps I am missing something? Can we approximate $f$ using some trigonometric polynomials different than partial sum of its Fourier series?

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3x89g2
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    I don't think you're missing anything. – Christopher A. Wong Apr 18 '16 at 23:31
  • @ChristopherA.Wong I am confused because I am wondering, is it true that for any $\epsilon>0$, $| f - S_N (f)| < \epsilon $ for all $x \in [-\pi,\pi]$, assuming $f$ is continuous? – 3x89g2 Apr 18 '16 at 23:41
  • OK, I see understand your concern now; it is my mistake. I misread your question. It's not the partial sum that's converging uniformly, but the Cesaro mean. It is false that the partial sums converge uniformly if $f$ is continuous. – Christopher A. Wong Apr 19 '16 at 01:26
  • @ChristopherA.Wong Yeah I realized that as well. That makes sense now because if the partial sum converges uniformly, then we just get $L^2$ convergence for free and there's no need to go through the long proof :p Guess I need some sleep. Thank you! – 3x89g2 Apr 19 '16 at 03:10

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