I'm trying to confirm or refute an intuition I have but I can't figure out exactly how to get a conclusive answer with my own calculations even though it is ostensibly fairly trivial.
Take the function:
$$f(x,y)=\mathrm{sinc}(x)~e^{-ix}+\mathrm{sinc}(y)~e^{-iy}\tag{1}$$
In the special case $x=-y$ we have:
$$f(x,y)={2}~\mathrm{sinc}(x)\cos(x)\tag{2}$$
Which makes me have an instinct that in the general case $x\neq{-y}$, we should have something like:
$$f(x,y)=\sum_{i,j}c_{ij}~\mathrm{sinc}(a_i(x,y))\times\cos(b_j(x,y))\tag{3}$$
But after several hours of trying to cleverly tease out what the form of $c_{ij}$, $a_i$, and $b_j$ are, I'm at a total loss. I still feel strongly that my instinct that there should be some form similar to this isn't totally wrong.
I suppose I'm just asking if anyone here has any insights to offer here.