指数多项式与从Fourier样本中识别多边形区域
Exponential polynomials and identification of polygonal regions from Fourier samples
AI总结：
本文研究了二元指数多项式的唯一确定性问题，并利用非构造性方法证明，仅需少量Fourier样本即可根据边缘斜率与顶点数等参数唯一识别多边形区域。
摘要：
考虑所有二元指数多项式的集合$E(D, N)$：$$ f(ξ, η) = \sum_{j=1}^n p_j(ξ, η) e^{2πi (x_jξ+y_jη)}, $$ 其中多项式$p_j \in \mathbb{C}[ξ, η]$的次数$<D$，$n\le N$，且$x_j, y_j \in \mathbb{T} = \mathbb{R}/\mathbb{Z}$。我们找到了一个仅依赖于$N$和$D$的集合$A \subseteq \mathbb{Z}^2$，其大小为$O(D^2 N \log N)$，使得$f$在$A$上的值能唯一确定$f$。注意，$A$的大小仅比写出$f$所需的参数数量多出一个对数级别的量。我们利用此结果证明了一些关于多边形区域的唯一性结论，这些结论基于其指示函数的Fourier变换的一小部分样本。如果多边形区域边缘的不同斜率数量$\le k$，则该区域可由一组预定的Fourier样本集确定，该集合仅依赖于$k$和最大顶点数$N$，且大小为$O(k^2 N \log N)$。在所有边缘均已知平行于坐标轴的特殊情况下，该多边形区域可由一组仅依赖于$N$且大小为$O(N \log N)$的Fourier样本集确定。我们的方法是非构造性的。
英文摘要：
Consider the set $E(D, N)$ of all bivariate exponential polynomials $$ f(ξ, η) = \sum_{j=1}^n p_j(ξ, η) e^{2πi (x_jξ+y_jη)}, $$ where the polynomials $p_j \in \mathbb{C}[ξ, η]$ 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.
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