let's define $f:\mathbb{R} \to \mathbb{C}$ a $2\pi$ periodic continuous function.
Let's say $s_{n}(x)=\sum_{k=-n}^{n} f\hat (k) e^{ikx}$.
Let's say $t_{n}(x)=\frac{1}{n+1}\sum_{k=0}^{n} s_{k}(x)$.
Prove that $t_{n}(x)=\sum_{k=-n}^{n} \frac{n-|k|+1}{n+1} f\hat (k) e^{ikx}$
my attempt: $t_{n}(x)=\frac{1}{n+1}\sum_{k=0}^{n} s_{k}(x) = \frac{1}{n+1}\sum_{k=0}^{n} \sum_{l=-k}^{k} f\hat (l) e^{ilx}$. Now I tried different things with the sommations and switch them, but i always got stuck with my $k,n,l$. Is there someone who can bekome $t_{n}(x)$ and explain me why you can switch the sommations?