It's well known that the Fourier series converges uniformly when a function is $C^2$ and periodic on say $[0,2\pi]$. If the function is not periodic you can have lack of uniform convergence near the endpoints due to "Gibbs" phenomenon. However I would like to understand if and when you still have uniform convergence on compact subsets of $[0,2\pi]$? This seems to be geometrically obvious but I can't find a good source for it. Any suggestions would be appreciated. For instance $f(x) = x$ on $[0,1]$. It would appear that the Fourier series converges uniformly on compact subsets of $[0,2\pi]$ but this doesn't seem to follow from analysis of the Fourier coefficients. Isn't it the same as if I had done an odd extension of $f(x)$ so that I have $-|x|$ on $[-2\pi,2\pi]$ and find the fourier series for this new domain? In the latter case I clearly get uniform convergence and I don't think this should depend on the fact that I reflected and chose a different domain.
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3$\begingroup$ I think you might want to restrict to compact subsets of $(0, 2\pi)$. In this case you can surely find your answer on the wikipedia page on convergence of Fourier series. $\endgroup$ – Matt Young Sep 30 '10 at 4:41
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$\begingroup$ When you say your function is non-periodic -- are you still assuming it is everywhere defined on the closed interval $[0,2\pi]$ and $C^2$ everywhere on that interval? $\endgroup$ – Yemon Choi Sep 30 '10 at 5:01
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$\begingroup$ The fourier series of the sawtooth wave function is (up to constants) $\sum_{n\ge 1} \sin (xn)/n$, but the triangle wave has the series $\sum_{n\ge 1} \cos (xn)/n^2$, which has a quite different rate of convergence than the former series. $\endgroup$ – Florian Sep 30 '10 at 13:09
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$\begingroup$ Yes $C^2$ up to the boundary on $[0,2]$, Yemon Choi. Florian: I think both of those series still converge uniformly though in the interior (the first one you show by using the Dirichlet series test). Matt Young: I did not see any reference to what you have mentioned actually. If I had found this I would have not asked the question here. $\endgroup$ – Dorian Sep 30 '10 at 14:16
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$\begingroup$ Then I don't understand what you mean by "Isn't it the same as if I had done ...". $\endgroup$ – Florian Sep 30 '10 at 14:29
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I assume you mean "...if and when you still have uniform convergence on compact subsets of (0,2π)? " This is in the nature of what is called a localization theorem. These go back to Riemann who proved that the convergence of the Fourier series of an $L^1$ function $f$ AT a point $x$ depends only on the behavior of $f$ in any small neighborhood of $x$. I'll stick my neck out a little and say that I wouldn't be too surprised if there is a generalization of Riemann's Localization Theorem that says that local uniform convergence near $x$ likewise only depends on the behavior of $f$ near $x$ (but that is pure conjecture on my part).
answered Sep 30 '10 at 4:50
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2$\begingroup$ Right, and the claim should follow already from the explicit form of the localization theorem, see springerlink.com/content/f4q3v26220670389 (link superseded by new link below). $\endgroup$ – Piero D'Ancona Sep 30 '10 at 10:17
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$\begingroup$ So this paper confirms the conjecture you've made? $\endgroup$ – Dorian Sep 30 '10 at 14:16
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$\begingroup$ Thanks for that reference, Piero. It is not exactly easy reading, but it does seem to confirm what I conjectured. $\endgroup$ – Dick Palais Sep 30 '10 at 21:28
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$\begingroup$ @PieroD'Ancona Would you happen to remember that reference? The Springer link is no longer working. $\endgroup$ – Todd Trimble♦ Jul 31 '20 at 18:42
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$\begingroup$ I suspect the paper was Riemann’s Localization Theorem. An Estimate for the Rate of Convergence, by S. A. Teliakovskii (Journal of Mathematical Sciences, Vol. 155, No. 1, 2008) available <a href="link.springer.com/content/pdf/10.1007/…> $\endgroup$ – Piero D'Ancona Aug 1 '20 at 20:19
