Computing a Fourier-Transform

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 Fourier-Transform 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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