锥面上的正体积压缩与最优超插值稳定性
Positive Cubature Compression and Optimal Hyperinterpolation Stability on a Conic Surface
AI总结：
本文通过保测的$\mathbb Z_2$商映射将锥面上的正体积转化为圆盘体积，实现节点压缩，并证明压缩后$L^2$条件数恒为1，$L^\infty$稳定性达到最优阶，从而降低采样成本。
AI中文摘要：
我们研究了在带Jacobi型权重的截断锥面上的正体积和超插值的计算成本与稳定性。采用经典二次圆盘到锥面的映射，以无基形式作为保测的$\mathbb Z_2$商。该映射将全次数$m$的锥迹空间等同于次数至多$2m$的偶圆盘多项式，并将正次数$2n$的锥体积转换为中心对称的正次数$4n+1$的圆盘体积，反之亦然。这实现了Möller下界及其节点超额的精确传递，因此近最小圆盘公式产生压缩的非乘积锥规则。同一商映射传递再生核和超插值算子。每个正次数$2n$的锥规则在$\Pi_n(V)$上给出精确的$L^2$采样等距；特别是，加权采样矩阵的条件数为1，与节点的数量和几何无关。对于无权重径向情形$\gamma=0$，若$\Lambda_n^V$表示次数$n$超插值的$C(V)\to C(V)$ Lebesgue常数，则每个正次数$2n$的锥体积规则满足与规则无关的尖锐律$$c n \le \Lambda_n^V \le C n.$$更一般地，当$\gamma$为非负半整数时，上界$\Lambda_{n,\gamma}^V \le C_\gamma n^{\gamma+1}$成立。因此，节点压缩在保持精确$L^2$条件数的同时，相关的$L^\infty$稳定性具有最优的普遍阶。低次数近最小规则和较高次数的优化圆盘方案说明了采样成本的降低。
英文摘要：
We study the computational cost and stability of positive cubature and hyperinterpolation on a truncated conic surface with a Jacobi-type weight. A classical quadratic disk-to-cone map is used in a basis-free form as a measure-preserving $\mathbb Z_2$ quotient. It identifies the full degree-$m$ conic trace space with the even disk polynomials of degree at most $2m$ and converts positive degree-$2n$ cone cubature into centrally symmetric positive degree-$(4n+1)$ disk cubature, and conversely. This yields an exact transfer of Möller's lower bound and of the node excess above it, so that near-minimal disk formulas produce compressed non-product cone rules. The same quotient transfers reproducing kernels and hyperinterpolation operators. Every positive degree-$2n$ cone rule gives an exact $L^2$ sampling isometry on $Π_n(V)$; in particular, the weighted sampling matrix has condition number one, independently of the number and geometry of the nodes. For the unweighted radial case $γ=0$, if $Λ_n^V$ denotes the $C(V)\to C(V)$ Lebesgue constant of degree-$n$ hyperinterpolation, then every positive degree-$2n$ cone cubature rule satisfies the rule-independent sharp law $$c n \le Λ_n^V \le C n.$$ More generally, the upper bound $Λ_{n,γ}^V \le C_γn^{γ+1}$ holds when $γ$ is a nonnegative half-integer. Thus node compression preserves exact $L^2$ conditioning while the associated $L^\infty$ stability has the optimal universal order. Low-degree near-minimal rules and higher-degree optimized disk schemes illustrate the reduction in sampling cost.
