Exponential polynomials and identification of polygonal regions from Fourier samples
<|reference_start|>Exponential polynomials and identification of polygonal regions from Fourier samples: Consider the set $E(D, N)$ of all bivariate exponential polynomials $$ f(\xi, \eta) = \sum_{j=1}^n p_j(\xi, \eta) e^{2\pi i (x_j\xi+y_j\eta)}, $$ where the polynomials $p_j \in \mathbb{C}[\xi, \eta]$ have degree $<D$, $n\le N$ and where $x_j, y_j \in \mathbb{T} = \mathbb{R}/\mathbb{Z}$. We find a set $A \subseteq \mathbb{Z}^2$ that depends on $N$ and $D$ only and is of size $O(D^2 N \log N)$ such that the values of $f$ on $A$ determine $f$. Notice that the size of $A$ is only larger by a logarithmic quantity than the number of parameters needed to write down $f$.   We use this in order to prove some uniqueness results about polygonal regions given a small set of samples of the Fourier Transform of their indicator function. If the number of different slopes of the edges of the polygonal region is $\le k$ then the region is determined from a predetermined set of Fourier samples that depends only on $k$ and the maximum number of vertices $N$ and is of size $O(k^2 N \log N)$. In the particular case where all edges are known to be parallel to the axes the polygonal region is determined from a set of $O(N \log N)$ Fourier samples that depends on $N$ only.   Our methods are non-constructive.<|reference_end|>
@article{kolountzakis2024exponential,
    title={Exponential polynomials and identification of polygonal regions from
  Fourier samples},
    author={Mihail N. Kolountzakis and Emmanuil Spyridakis},
    journal={arXiv preprint arXiv:2409.01432},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.01432},
    primaryClass={math.CA cs.NA math.MG math.NA}
}