Let $a \in (0,1/2]$ and define $ f:\mathbb T\rightarrow \mathbb R $ by $$ f(x) = \begin{cases} 1, &\text{if $x$ is between $-a$ and $a$} \\ 0, &\text{otherwise} \end{cases} $$

I figured out all the Fourier coefficients of $f$, but the question asks me to show the sum of all these Fourier coefficients converges as an infinite series to $f(0)$. Could someone help me with this problem? Thank you so much.

  • $\begingroup$ What kind of theorems do you know that guarantee convergence of Fourier series? $\endgroup$ – mrf Oct 4 '15 at 22:18

You can use Dini's test in this case, since the function is pretty simple. MathWorld

  • $\begingroup$ From what I know, Dini's test proves uniform convergence of a monotone sequence of continuous functions converging pointwise to a continuous function on a compact set. But now the problem is about the convergence of a numerical series. So could you elaborate on what you mean by using Dini's test to show the sum of all the Fourier coefficients converges to f(0)? $\endgroup$ – thomas Oct 5 '15 at 11:28

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