多元三角多项式的正求积与移动采样
Positive quadrature and mobile sampling of multivariate trigonometric polynomials
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中文总结 AI 辅助
本文研究多元三角多项式的正广义求积，推导其覆盖半径上界，对比求积曲线下界方法，证明秩1 Korobov格规则曲线为准最优且适配2-球面。
中文摘要 AI 辅助
一维及多维求积点的几何性质是数值分析的经典课题。近年来，用沿曲线的积分替代离散点与权重的广义求积方法受到越来越多的关注。本文聚焦对多元三角多项式精确的正广义求积，利用符号局部化测试函数推导这类求积的覆盖半径上界，该技术也可自然扩展至区间上的代数多项式、球多项式及双曲交叉三角多项式。对于求积曲线长度的下界，本文在多元三角多项式框架下对比两种近期方法。第二部分研究周期函数秩1 Korobov格规则所基于的几何结构简单的知名曲线，证明这些曲线在多个最优性准则下为准最优，并展示其对2-球面的适应性。
英文摘要
The geometric properties of quadrature points in one and multiple dimensions are a classical topic in numerical analysis. Recently, generalized quadrature methods, where discrete points and weights are replaced by integration along curves, have attracted growing interest. This paper focuses on positive generalized quadratures that are exact for multivariate trigonometric polynomials. We derive upper bounds on the covering radius of such quadratures using sign-localized test functions, a technique that also extends naturally to algebraic polynomials on intervals, spherical polynomials, and hyperbolic cross trigonometric polynomials. For lower bounds on the length of quadrature curves, we compare two recent approaches in the context of multivariate trigonometric polynomials. The second part of the paper studies a well-known, geometrically simple curve that underlies rank-1 Korobov lattice rules for periodic functions. We prove that these curves are quasi-optimal with respect to multiple optimality criteria and demonstrate its adaptability to the 2-sphere.
