傅里叶比：关于紧支撑测度的不确定性、限制与近似
The Fourier Ratio: Uncertainty, Restriction, and Approximation for Compactly Supported Measures
AI总结：
本文提出傅里叶比概念，用于研究紧支撑测度的不确定性、限制与近似问题，揭示了信号恢复与分形测度之间的差异。
摘要：
我们引入了紧支撑贝塞尔测度的连续傅里叶比。对于在\mathbb{R}^d上定义的测度μ和f∈L^2(μ)，傅里叶比比较正则化傅里叶变换在尺度R下的L^1和L^2范数。我们开发了分形不确定性原理，给出了空间和频率支撑的覆盖数的精确双侧界限，应用于精确信号恢复。我们证明小傅里叶比意味着在L^1、L^2和L^∞中能通过低次三角多项式高效近似。相反，限制估计揭示了曲面测度与随机分形测度之间的显著差距，从而得出对近似次数的强下界。还获得了凸曲面测度的应用。
英文摘要：
We introduce a continuous analog of the Fourier ratio for compactly supported Borel measures. For a measure \(μ\) on \(\mathbb{R}^d\) and \(f\in L^2(μ)\), the Fourier ratio compares \(L^1\) and \(L^2\) norms of a regularized Fourier transform at scale \(R\). We develop a fractal uncertainty principle giving sharp two-sided bounds in terms of covering numbers of spatial and frequency supports, with applications to exact signal recovery. We show that small Fourier ratio implies efficient approximation by low-degree trigonometric polynomials in \(L^1\), \(L^2\), and \(L^\infty\). In contrast, restriction estimates reveal a sharp gap between curved measures and random fractal measures, yielding strong lower bounds on approximation degree. Applications to convex surface measures are also obtained.
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