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?
Asked
Active
Viewed 44 times
-2
-
About usage: "Fourier transform of function" is usual, in your situation above. – paul garrett Oct 02 '14 at 21:21
1 Answers
2
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
$$ g(k_x,k_y) = \bigg(\int_{-1}^1 e^{ik_x x}\,dx\bigg)\bigg(\int_{-1}^1 e^{ik_y y}\,dy\bigg) = \frac{4 \sin (k_x) \sin (k_y)}{k_x k_y}. $$
ybeltukov
- 296
-
An obvious but useful sanity check: The limit as $(k_x,k_y)\to(0,0)$ is just $4$ which is the same as the area enclosed in the $xy$-plane. – Semiclassical Oct 02 '14 at 21:20