In the main theorem of Fourier Series, it converge to the mean of $f$ around $x$,which is $\frac{1}{2}\left[f(x+0)+f(x-0)\right]$,with condition that both $f$, $f'$ and $f''$ are sectionally continuous。 And in its corollaries, Fourier series converge to $f(x)$ with relaxed condition that is only $f$ and $f'$ are sectionally continuous. What's the difference? And are there any applications of the difference?
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do you see this is about the convolution with the Dirichlet kernel ? do you see any obvious condition for $f \ast D_n\to f$ uniformly ? – reuns Aug 31 '16 at 04:22
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I'v not learned the definition of convolution yet.So I need a sample as your mentioned about.@user1952009 – NFDream Aug 31 '16 at 06:27