Computing a FourierTransform
We have $M:= \left \{ \left(\begin{array}{c} x \\ y \end{array} \right ) \in \mathbb{R}^2 ; \left  \left  \left(\begin{array}{c} x \\ y \end{array} \right )  \left(\begin{array}{c} 0 \\ \frac{1}{\sqrt{2} } \end{array} \right ) \right  \right _1 \leq \frac{1}{\sqrt{2} } \right \}$.
Compute the FourierTransform of the charactersitic function of $M$, i.e.
$\chi_M : \mathbb{R}^2 \rightarrow \mathbb{R}, \quad v \mapsto \begin{cases} 1 & v \in M \\ 0 & v \notin M \end{cases}$.
Do that by using the following hint:
$\textbf{Hint}$: Consider the matrix $A= \frac{1}{\sqrt{2} } \begin{pmatrix} 1 & 1 \\ 1 & 1\end{pmatrix}$ first and determine the set $AM = \{ A \cdot v ; v \in M \}$.
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are you able to put it in latex? if not, I can copy the question and afterwards accept the answer here

I've tried to rewrite my solution that was deleted rather hastily  if there's any problem or typo let me know. Sorry for the inconvenience.
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