Prove the classical full Fourier series of $f(x)$ converges uniformly to $f(x)$ if $f(x)$ is continuous of period $2\pi$ and its derivative $f'(x)$ is piecewise continuous.
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Do you know "Dini's Test"? – David C. Ullrich Apr 05 '16 at 16:44
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The idea is to write down the error at a given point in terms of an infinite series of integrals, then integrate by parts (moving the derivative to $f $). For high frequencies this introduces a factor which goes to zero as the frequency goes to infinity. Use that factor and Parseval's identity to get an error bound independent of $x$. (The bound should be something like $| f' |_{L^2}/N $). – Ian Apr 05 '16 at 16:47
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For more info on this technique, look up "spectral convergence"; this is the basis of spectral methods for a class of problems whose solution is smooth. – Ian Apr 05 '16 at 16:50
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@Ian Seems to me if we want that sort of argument it could be stated more simply: Integration by parts shows that $\widehat{f'}(n)=in\hat f(n)$. And $f'\in L^2$, so $\sum n^2|\hat f(n)|^2<\infty$. Hence $\sum|\hat f(n)|<\infty$, so the series converges uniformly. – David C. Ullrich Apr 05 '16 at 17:05
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@DavidC.Ullrich While that is simpler, expanding on it a bit hints at the fact that we can keep integrating by parts for sufficiently smooth $f$ to get better and better estimates. – Ian Apr 05 '16 at 20:46