How hard is it to estimate a discrete-time signal $(x_{1}, ..., x_{n}) \in \mathbb{C}^n$ satisfying an unknown linear recurrence relation of order $s$ and observed in i.i.d. complex Gaussian noise? The class of all such signals is parametric but extremely rich: it contains all exponential polynomials over $\mathbb{C}$ with total degree $s$, including harmonic oscillations with $s$ arbitrary frequencies. Geometrically, this class corresponds to the projection onto $\mathbb{C}^{n}$ of the union of all shift-invariant subspaces of $\smash{\mathbb{C}^\mathbb{Z}}$ of dimension $s$. We show that the statistical complexity of this class, as measured by the squared minimax radius of the $(1-δ)$-confidence $\ell_2$-ball, is nearly the same as for the class of $s$-sparse signals, namely $\smash{O\left(s\log(en) + \log(δ^{-1})\right) \cdot \log^2(es) \cdot \log(en/s).}$ Moreover, the corresponding near-minimax estimator is tractable, and it can be used to build a test statistic with a near-minimax detection threshold in the associated detection problem. These statistical results rely upon a simple analytic observation: the interpretation of the Fourier coefficients of the Christoffel function of any shift-invariant subspace of $\smash{\mathbb{C}^\mathbb{Z}}$ as a reproducing filter with the smallest possible spectrum in all $\ell_p$-norms, $p \in [1,\infty]$, at once.

翻译：估计一个满足未知$s$阶线性递归关系并在独立同分布复高斯噪声中观测到的离散时间信号$(x_{1}, ..., x_{n}) \in \mathbb{C}^n$有多困难？所有此类信号的类虽为参数化，但极其丰富：它包含$\mathbb{C}$上所有总次数为$s$的指数多项式，包括具有$s$个任意频率的谐波振荡。几何上，该类对应于$\smash{\mathbb{C}^\mathbb{Z}}$中所有维度为$s$的移不变子空间之并在$\mathbb{C}^{n}$上的投影。我们证明，以$(1-δ)$置信度$\ell_2$球的平方极小极大半径衡量，该类的统计复杂度与$s$-稀疏信号类几乎相同，即$\smash{O\left(s\log(en) + \log(δ^{-1})\right) \cdot \log^2(es) \cdot \log(en/s).}$ 此外，相应的近极小极大估计器是可处理的，并可用来在关联的检测问题中构建具有近极小极大检测阈值的检验统计量。这些统计结果依赖于一个简单的分析观察：将$\smash{\mathbb{C}^\mathbb{Z}}$中任意移不变子空间的Christoffel函数的傅里叶系数解释为在所有$\ell_p$-范数（$p \in [1,\infty]$）下同时具有最小可能频谱的再生滤波器。