Exam

Let

n

be an even integer. Find the Fourier-transform of the function

f : C_{n v} \to C

f = δ_{σ}

, i.e.

f (σ) = 1

and

f (x) = 0

for all

x \neq σ

$\hat{f} (χ_{1}) = 1 / 2 n$ , $\hat{f} (χ_{2}) = - 1 / 2 n$ , $\hat{f} (χ_{3}) = 1 / 2 n$ , $\hat{f} (χ_{4}) = - 1 / 2 n$ and for $j = 1, \dots, n / 2 - 1$ : $\hat{f} (ψ_{j}) = \frac{1}{2 n} (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}) .$ Thus the Fourier inversion formula yields: $f (x) = \frac{1}{2 n} (χ_{1} (x) - χ_{2} (x) + χ_{3} (x) - χ_{4} (x)) + \frac{1}{n} \sum_{j = 1}^{n / 2 - 1} t r ((\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}) Ψ_{j} (x))$