TITLE: Show the Fourier series converges pointwise to $f$, without using the fourier coefficients.
QUESTION [0 upvotes]: Consider the Fourier series of the function $f(x)= \frac{\pi}{4}|sin(x)|$.

Show the Fourier series converges pointwise to $f$
Calculate the Fourier coefficients of $f$
Prove that the Fourier series converges uniformly on  $\mathbb{R}$ to $f$


I'm stuck on trying to prove that the Fourier series converges pointwise, without calculating the coefficients first.
I did manage to do the last two questions, but I don't know how I can start the first one without using the coefficients.

REPLY [1 votes]: It can be proved, that Fourier series $S(x)$ of piecewise smooth function on $(-\pi, \pi)$ converges pointwise to the following value:
$$
\begin{aligned}
S(x) &= \frac{f(x+0)+f(x-0)}{2},\ \forall x \in (-\pi, \pi)\\
S(-\pi) = S(\pi) &= \frac{f(\pi-0)+f(-\pi+0)}{2}
\end{aligned}
$$
Fortunately, $f(x+0) = f(x-0) = f(x),\ \forall x \in (-\pi, \pi)$, so that
$$
S(x) = f(x), \forall x \in (-\pi, \pi)
$$
Since $f(\pi-0) = f(-\pi+0) = 0$
$$
\begin{aligned}
S(\pi) &=  0 = f(\pi)\\
S(-\pi) &= 0 = f(-\pi)
\end{aligned}
$$
Because of periodicity
$$
S(x) = f(x),\ \forall x \in R
$$
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