Approximation of smooth real functions with an integer integral by a sum of $\delta$-functions with coefficients $\pm 1$

Question

This question arose from a quantum physics research (my article Entropy 2022, 24(2), 261, where I considered approximation of a smooth charge density (built from a wave function) by a collection of point-like particles and antiparticles with quantized charge).

Let us consider a smooth real function $f(\boldsymbol{x})$ in a $d$-dimensional cube $C$:$-\pi\leq x^i\leq\pi$, where $\boldsymbol{x}=(x^1,\ldots,x^d)$, $1\leq i\leq d$, $\int\limits_C f(\boldsymbol{x})\boldsymbol{dx}=1$, and $f(\boldsymbol{x})$ vanishes on the boundary of the cube.

Let us also consider distributions of the form $$g(\boldsymbol{x})=\sum_{n=1}^{2j+1}(-1)^{n+1}\delta(\boldsymbol{x}-\boldsymbol{x}_n),$$where $\delta(\boldsymbol{x})$ is the $d$-dimensional Dirac's $\delta$-function, and $\boldsymbol{x}_n$ belong to the interior of $C$.

$f(\boldsymbol{x})$ and $g(\boldsymbol{x})$ in the cube can be expanded in Fourier series with terms of the form $a_\boldsymbol{k}\exp{(i\boldsymbol{k}\boldsymbol{x}})$, where $d$-dimensional vectors $\boldsymbol{k}=(k^1,\ldots,k^d)$ have integer coordinates (or the series can involve sines and cosines). For some natural $m$, the series can be truncated by keeping only terms with such $\boldsymbol{k}$ that $|k^i|\leq m$.

I suspect that the following statement is true:

$\forall m$ $\exists$ $j$ and $\boldsymbol{x}_n$ ($1\leq n\leq 2j+1$) such that the truncated Fourier series for $f(\boldsymbol{x})$ and $g(\boldsymbol{x})$ coincide.

The reason to think so is that, on the one hand, the coincidence of the truncated series is equivalent to existence of roots of some polynomial system, on the other hand, the number of the unknowns of the system can be arbitrarily large. However, it is not obvious that real solutions for $\boldsymbol{x}_n$ can be found for a sufficiently large $j$. I was only able to prove the statement for $d=1$ (my article Quantum Rep. 2022, 4(4), 486-508, Section 2.8). However, I was not able to prove the statement for $d>1$ because of difficulties with vectors $\boldsymbol{k}$ that have some zero coordinates.

I would much appreciate some ideas of the proof (preferably constructive) for $d>1$ (say, $d=3$). Besides, I cannot be sure that similar problems have not been considered by others, so I would also be grateful for references.

Answer 1

I can do the easy part (to show how to prove the statement you need) but I'll have to leave the difficult part (to find out who, when, and where obtained this result for the first time) to someone else. The argument is based on my AoPS post from long ago, so you'll have to read and understand it first before reading the rest of this answer. Once that is done, the rest of the argument is as follows:

We put one delta-measure wherever you want to get the $0$-th coefficient right and the rest will be composed of dipoles $\delta(x-a)-\delta(x-b)$. Let $K$ be the set of all sequences of the Fourier coefficients of interest with respect to the $\sin/\cos$ system (to keep everything within the real domain) of such dipoles when $a,b$ run over $[-\pi,\pi]^d$ (you can go slightly inside, if you want, though how much depends on how many Fourier coefficients you want to handle). Now the diameter of $K$ in the corresponding finite dimensional Euclidean space is some fixed quantity and, most importantly, $K$ is connected. On the other hand, the convex hull of $K$ contains a small ball (if you have any trigonometric polynomial including only non-zero frequencies and of $L^2$ norm $1$, then we can find a dipole with respect to which its integral is at least $1$). Hence the convex hull of the Minkowski sum $K+\dots+K$ contains as large ball as we wish and, thereby, the sum itself contains as large ball as we wish, provided that the number of summands is large enough. The End.

P.S. My initial application of that lemma was completely different and I managed to bypass it before I found its proof, so it was just hanging in the air all these years. It is funny that it turned out to be useful now. Also I wouldn't call this proof "constructive" but I strongly suspect that some high-dimensional analogue of the intermediate value theorem has to be involved (I essentially used Brouwers fixed point theorem), so, perhaps, one cannot do much better. I will be happy to get proved wrong on that account, of course :-)