Let $f$ a complex function, $f \in L^2[-\pi,\pi]$. Let $c_n=\frac{1}{2\pi} \int_{-\pi}^{\pi}f(x)e^{-inx}dx$ the Fourier coefficients, $n \in \mathbb{Z}$.
Problem 1. Find fourier coefficients $\{c_n(h)\}_{n \in \mathbb{Z}}$ of the Steklov function $$f_h(x)=\frac{1}{2h}\int_{x-h}^{x+h}f(t)dt, x\in [-\pi,\pi]$$ where $f$ is extended throughout $\mathbb{R}$ as a $2\pi$-periodic function.
Problem 2 Apply Parseval equality, to show that $f_h \in L^2[-\pi,\pi]$
Thanks to this problem I proved that $$c_n(f_h)=\frac{c_n}{hn}sin(hn)$$. But for the second problem, the equality Parseval is $\sum_{n \in \mathbb{Z}}|c_n(f_h)|^2=||f_h||_{L^2}^2$. So $$\sum_{n \in \mathbb{Z}}|c_n(f_h)|^2=\sum_{n \in \mathbb{Z}}|c_n|^2*(\frac{sin(hn)}{hn})^2$$ we know that $\sum_{n \in \mathbb{Z}}|c_n|^2$ converges and $\frac{sin^2(hn)}{h^2n^2} \leq \frac{1}{h^2}$, so $\{\frac{sin^2(hn)}{h^2n^2}\}$ is bounded and if we prove that $\{\frac{sin^2(hn)}{h^2n^2}\}$ is monotonic, by Abel criterion $\sum_{n \in \mathbb{Z}}|c_n(f_h)|^2$ converges and we finish. So could you help me to prove the last thing? or do you know another form to solve it?, thank you.