双曲交叉三角多项式一致范数的采样离散化
Sampling discretization of the uniform norm for hyperbolic-cross trigonometric polynomials
- University of Alberta（阿尔伯塔大学）
- University of Manitoba（曼尼托巴大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究双曲交叉三角多项式一致范数的采样离散化，构造显式赋范集并证明随机采样达到最优指数，同时给出赋范集大小的下界，排除二次界。
AI中文摘要：
我们研究了在$\T^d$上频率位于水平$N$的双曲交叉中的三角多项式的一致范数的采样离散化问题。对于每个$d,N\ge2$和$\varepsilon\in(0,1)$，我们构造了一个显式的赋范集，其点数至多为$\bigl(C_1(1+\varepsilon^{-1})\bigr)^{d-1}N^{1+\varepsilon}$，赋范常数至多为$\bigl(C_2(1+\varepsilon^{-1})\bigr)^{d-1}$，其中$C_1,C_2$是绝对常数。我们还证明了独立的Haar分布点能达到相同的指数$1+\varepsilon$：大小为至少$C(d,\varepsilon)N^{1+\varepsilon}\log(2/\eta)$的样本以至少$1-\eta$的概率同时赋范整个空间，赋范常数以$\exp(C_d\varepsilon^{-d})$为界且与$N$无关。确定性构造结合了均匀稳定的de la Vallée Poussin采样算子与Smolyak型组合恒等式。随机结果则来自一个关于与交换正则划分相关的空间之和的抽象赋范定理，以及双曲交叉多项式的多尺度逼近。最后，我们证明对于频率在$\{-1,0,1\}^d$中的三角多项式，任何常数$B>1$的赋范集至少有$\bigl(\pi d/(2e\log B)\bigr)^{d/2}$个点。这排除了在赋范常数固定时，多项式空间维度的二次界且其前因子仅随$d$多项式增长的可能性。
英文摘要：
We study sampling discretization of the uniform norm for trigonometric polynomials with frequencies in a hyperbolic cross of level $N$ on $\T^d$. For every $d,N\ge2$ and $\varepsilon\in(0,1)$, we construct an explicit norming set with at most $\bigl(C_1(1+\varepsilon^{-1})\bigr)^{d-1}N^{1+\varepsilon}$ points and norming constant at most $\bigl(C_2(1+\varepsilon^{-1})\bigr)^{d-1}$, where $C_1,C_2$ are absolute constants. We also prove that independent Haar-distributed points achieve the same exponent $1+\varepsilon$: a sample of size at least $C(d,\varepsilon)N^{1+\varepsilon}\log(2/η)$ norms the entire space simultaneously with probability at least $1-η$, with a norming constant bounded by $\exp(C_d\varepsilon^{-d})$ and independent of $N$. The deterministic construction combines uniformly stable de la Vallée Poussin sampling operators with a Smolyak-type combination identity. The random result follows from an abstract norming theorem for sums of spaces associated with commuting regular partitions, together with a multiscale approximation of hyperbolic-cross polynomials. Finally, we show that every norming set of constant $B>1$ for the trigonometric polynomials with frequencies in $\{-1,0,1\}^d$ has at least $\bigl(πd/(2e\log B)\bigr)^{d/2}$ points. This rules out quadratic bounds in the dimension of the polynomial space with a prefactor growing only polynomially in $d$ when the norming constant is fixed.
