I am trying to evaluate the following expression $$\mathcal{F}\{\mathrm{rect}_{L_{x}}(x)\mathrm{rect}_{L_{y}}(y)\}$$ which denotes the 2-dimensional Fourier transform (reciprocal variables $k_x$, $k_y$) of a product of two rectangular functions defined by $$\mathrm{rect}_{L_{x}}(x)=\begin{cases}1, & -L_x<x<L_x \\ 0, & \text{otherwise}\end{cases}$$
This is what I tried so far ("*" denotes convolution): $$\mathcal{F}\{\mathrm{rect}_{L_{x}}(x)\mathrm{rect}_{L_{y}}(y)\} = \mathcal{F}\{\mathrm{rect}_{L_{x}}(x)\}*\mathcal{F}\{\mathrm{rect}_{L_{y}}(y)\}\\ ={L_xL_y\over\pi^2}\mathrm{sinc}(L_xk_x)*\mathrm{sinc}(L_yk_y)\\ ={L_xL_y\over\pi^2}\int_{-\infty}^{\infty}\mathrm{d}k_x'\,\mathrm{sinc}(L_xk_x)\int_{-\infty}^{\infty}\mathrm{d}k_y'\mathrm{sinc}(L_y(k_y-k_y')) $$ Now both these integrals are Dirichlet integrals and evaluate to ${\pi \over L_x}$ and ${\pi \over L_y}$ respectively which yields total result of $1$.
I suspect there's a mistake somewhere, since this is not the result I am expecting to see. Moreover when I try to do the calculation in Mathematica
FourierTransform[UnitBox[x] UnitBox[y], {x, y}, {a, b}]
I get $${\mathrm{sinc}(a/2)\mathrm{sinc}(b/2)\over 2\pi}$$ which makes much more sense for the bigger problem I am solving.
Where am I wrong?