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Let $f$ and $g$ be continuous functions of period $2\pi$ and let $f\leq g$. I wonder the conditions on $f,g$ which guarantee the ineqality $FS(f)\leq FS(g)$, where $FS$ denote Fourier series of a function.

guest
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  • If $f$ and $g$ are continuous... no problem... – Jean Marie Feb 04 '18 at 19:41
  • Thanks for your comment. Is there any reference or proof available? – guest Feb 04 '18 at 19:53
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    $f$ continuous $\implies f=FS(f)$, plainly. See the main convergence theorem on Fourier series. Problems arise when you have jumps... and I have some doubt there exists theorems warranting that $f<g \implies FS(f) < FS(g)$ – Jean Marie Feb 04 '18 at 19:54
  • The problem is with Gibbs phenomenon (you must have heard about it) at discontinuity points. – Jean Marie Feb 04 '18 at 19:59
  • In Stein's book(Fourier Analysis : An Introduction, page 83), example of a continuous function whose Fourier series does not converge is given. – guest Feb 04 '18 at 20:07
  • By continuous, I mean continuous on whole $\mathbb{R}$. If your function $f$ is build by periodizing a function $\varphi$ defined on $[0,2 \pi]$ but with $\varphi(0) \neq \varphi(2 \pi)$, you have a jump and many things can arrive... – Jean Marie Feb 04 '18 at 20:11
  • I am a bit confused by these: – guest Feb 04 '18 at 21:02
  • https://math.stackexchange.com/questions/77855/pointwise-convergence-of-fourier-series – guest Feb 04 '18 at 21:03
  • I beg your pardon, I was wrong, we have to assume $f \in C^1$ for warranting $f=FS(f)$. $f \in C^0$ isn't enough. – Jean Marie Feb 04 '18 at 21:10
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    not problem, thanks for your valuable helps – guest Feb 04 '18 at 21:30

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