Localization in fourier series says.
At point $x_0$ behavior of fourier series of function $f$ is dependent on values of $f$ on any small neighbourhood of point $x_0$.
Proof is like this.
Pick $\delta \in (0,\pi)$ and partition Dirichlet integral into two parts.
$f(x_0-t)$ is left limit at point $x_0$
Now book uses Riemann lemma so I will write here what it says.
If $g$ is Riemann integrable on $[a,b]$ then
$lim_{\lambda \to \infty}\int_a ^b g(t)sin(\lambda t)dt=0$
Continuing proof it says that second sum of Dirichlet integral from Riemann lemma approaches $0$ when $n\to \infty$.
Can you help me understand this part? If we want to use Riemann lemma we must show that $\frac{f(x_0-t)+f(x_0+t)}{2sin{\frac{t}{2}}}$ is integrable on $[\delta,\pi]$ yes?
Because we are writing fourier series of $f$ that means $f$ is integrable on $[-\pi,\pi]$ and has $2\pi$ period.So $f(x_0-t)+f(x_0+t)$ is bounded. $\frac{f(x_0-t)+f(x_0+t)}{2sin(\frac{t}{2})}$ is also bounded because for example $2sin(\frac{t}{2})\geq-2$ so from bounded function on segment is continuous so is also integrable. Is my thinking correct?