Hi, friends! Let ##f:[a,b]\to\mathbb{C}## be an http://librarum.org/book/10022/173 [Broken] periodic function and let its derivative be Lebesgue square-integrable ##f'\in L^2[a,b]##. I have read a proof (p. 413 here) by Kolmogorov and Fomin of the fact that its Fourier series uniformly converges to a continuous function ##\varphi## whose Fourier coefficients are the same as the Fourier coefficients of ##f##.(adsbygoogle = window.adsbygoogle || []).push({});

I read in the same proof that, since ##\varphi##has the same Fourier coefficients of##f##,because of the continuity of the two functions we get##f=\varphi##. I do not understand why continuity guarantees the equality. Could anybody explain that?

I ##\infty##-ly thank you!!!

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# Identical Fourier coefficients of continuous ##f,\varphi\Rightarrow f=\varphi##

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