$L_2$ 型 Marcinkiewicz-Zygmund 不等式所需的点数量
Required Number of Points in $L_2$ Marcinkiewicz-Zygmund Inequalities
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中文总结 AI 辅助
本文确定了 $m$ 维复函数空间加权 $L_2$ 型 Marcinkiewicz-Zygmund 不等式所需点数量的渐近阶，构造了难离散化的函数空间得到匹配下界，并推导了加权最小二乘方程组的相关条件结论。
中文摘要 AI 辅助
我们确定了 $m$ 维复函数空间的加权 $L_2$ 型 Marcinkiewicz-Zygmund 不等式在最坏情况下所需的点评估数量，其结果在绝对常数范围内有效。若相对失真度为 $0<\varepsilon<1$，则该数量为 $\Theta\Big(\min\Big\{m^2,\frac{m}{\varepsilon^2}\Big\}\Big)$，且精确离散化的最坏情况值为 $m^2$。上界由近期的构造得出，而本文的贡献在于构造了难以离散化的函数空间，得到了匹配的下界。我们采用了单位范数紧框架的加权子框架的迹方差不等式，其中完全图边框架便是一个实例，可在任意维度上构造；当 $m-1$ 为素数幂时，Singer 等角紧框架可改进常数；当最大等角紧框架存在时，其能给出本文方法可实现的最强界。此外，我们还推导了加权最小二乘方程组的条件数，以及用 LSQR 求解此类方程组时基于标准条件数的迭代估计的相关结论。
英文摘要
We determine, up to absolute constants, the worst-case number of point evaluations required for a weighted $L_2$ Marcinkiewicz-Zygmund inequality for an $m$-dimensional complex function space. If $0<\varepsilon<1$ is the relative distortion, this number is $$Θ\Big(\min\Big\{m^2,\frac{m}{\varepsilon^2}\Big\}\Big),$$ and exact discretization has the sharp worst-case value $m^2$. While the upper bounds follow from recent constructions, our contribution is the construction of function spaces that are hard to discretize and yield matching lower bounds. We use a trace-variance inequality for weighted subframes of unit-norm tight frames. One such instance is the complete-graph edge frame, which yields a construction in every dimension. Singer equiangular tight frames improve the constant when $m-1$ is a prime power, while maximal equiangular tight frames give the strongest bound possible using our method whenever they exist. We also derive consequences for the conditioning of weighted least-squares systems and for standard condition-number-based iteration estimates when these systems are solved by LSQR.
