Fourier and harmonic analysis

Let $f(x) = 3 + 4\cos(2x) - 2\sin(5x)$ on $\mathbb T$. Compute $\displaystyle \sum_{n \in \Z} |\hat f(n)|^2$, with $\hat f(n) = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x) e^{-inx}\,dx$.

Which property of an approximate identity does the Dirichlet kernel $D_N$ fail, and why does that matter?

For the square wave $f = \operatorname{sign}(\sin x)$, what happens to $\max_x S_N f(x)$ as $N \to \infty$?

The function $f(x) = e^{-2\pi|x|}$ has Fourier transform $\hat f(\xi) = \dfrac{1}{\pi(1 + \xi^2)}$, and Poisson summation holds for it. What is $\displaystyle\sum_{k \in \Z} \frac{1}{1 + k^2}$?

In the proof of the Heisenberg inequality we reach $\int|f|^2 \le 2\left(\int x^2|f|^2\right)^{1/2}\left(\int|f'|^2\right)^{1/2}$. Which result converts $\int |f'|^2$ into a statement about $\hat f$?

Which Lean statement correctly expresses the orthogonality of the characters of $\Z/N$ as "the sum of the powers of a non-trivial $N$-th root of unity is $0$"?

Let $f \in L^1(\R)$. Which statements are always true? Select all that apply.

Write dft(a) returning the discrete Fourier transform $A_k = \sum_{j=0}^{N-1} a_j e^{-2\pi i jk/N}$, $k = 0, \dots, N-1$, of a list a of $N$ numbers (use cmath). The tests check Parseval's identity for $\Z/N$: $\sum_k |A_k|^2 = N\sum_j |a_j|^2$.