I have a functional map from $(x,y) \in \mathbb{R}^2$ to another function $f_{x,y}(z,w) \in \mathbb{C}$. (Variables $z,w $ range from $-\infty$ to $\infty$.) That is, for any pair $x,y$, I get a complex-valued function that has domain $\mathbb{R}^2$. For my set of functions $f_{x,y}(z,w)$, I managed to prove $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{x,y}(z,w)f_{x+\delta_x,y+\delta_y}^\ast(z,w)\mathrm{d}z\mathrm{d}w = \mathrm{sinc}\left(\sqrt{\delta_x^2+\delta_y^2}\right),$$ (where $\mathrm{sinc}(1)=0$).
I now wonder about a basis for such signals, in particular its size. Let $(x,y) \in \mathcal{A}\subset \mathcal{R}^2$. Then I would like to express $$f_{x,y}(z,w)=\sum_{\ell}^L a_{\ell}(x,y) \phi_{\ell}(z,w)$$ for some basis functions $\phi_{\ell}(z,w)$ and expansion coefficients. What would be the "density" of basis functions, $$\lim_{|\mathcal{A}|\to \infty} \frac{1}{|\mathcal{A}|}L$$ required to capture (close to) all of the energy in $f_{x,y}(z,w)$ ? That is, how many basis functions are needed per $m^2$ (assuming that $(x,y)$ are coordinates in physical space) ?
I have a feeling that this is related to prolate spheroidal wave functions, but never worked with 2D-pswfs
