Fourier transform of following equation

Question

Say I have the following equation

$$ f(x,y) = \left\{ \begin{array}{lr} 1 & \text{if} \;|x|,|y| \leq 1 \\ 0 & \text{otherwise} \end{array} \right. $$

What is the Fourier transform of this equation?

Answer 1

The conditions $|x|,|y|<1$ mean the integration from $-1$ to $1$ over $x$ and $y$ in the following integral $$ g(k_x,k_y) = \int_{-1}^1\int_{-1}^1 e^{ik_x x+ik_y y}\,dx\,dy. $$

This integral is a multiplication of two simple 1-dimensional integrals

$$ g(k_x,k_y) = \bigg(\int_{-1}^1 e^{ik_x x}\,dx\bigg)\bigg(\int_{-1}^1 e^{ik_y y}\,dy\bigg) = \frac{4 \sin (k_x) \sin (k_y)}{k_x k_y}. $$