Say I have a function that is equal to 1 at two unit area squares. One is centered at $(-3,0)$ and the other at $(3,0)$. I am trying to find a formula for this function using only the rect function and the $\delta$ (impulse) function. I am then trying to find the Fourier transform of this function. Here is my attempt:
$$f(x,y) = \mathrm{rect}(x,y) * \delta(x-3,y)+ \mathrm{rect}(x,y) * \delta(x+3,y)$$
$$F(u,v) = \mathrm{sinc}(\pi u, \pi v)\,e^{-6 i\pi u} + \mathrm{sinc}(\pi u, \pi v)\,e^{6 i\pi u}$$
Addition Information: The rect function is simply equal to $1$ at the unit square centered at the origin and $0$ elsewhere. Convolving with the shifted Dirac delta function should shift the rect function from my understanding.