Suppose we have a function $f:\mathbb{R}^d\to\mathbb{R}$. Using the notation of https://www.maths.unsw.edu.au/sites/default/files/amr08_5_0.pdf, which defines an order $q$ function $g:\mathbb{R}^d\to\mathbb{R}$ as a function that can be written as a sum of terms, each of which only depend on $q<d$ of the variables. My question is if there is any existing work that provides a bound (preferably something like $\ell_\infty$) on the best order $q$ approximation to an arbitrary $f$.
This is extremely broad, so I’m not expecting much. In particular, I’m interested in functions $f$ that satisfy the conditions for the Fourier inversion theorem, i.e., are absolutely integral and continuous.
To make the problem a bit simpler, suppose $f$ is already of order $p$. Then what about the best order $q<p$ approximation of $f$?