Marcinkiewicz-Zygmund不等式用于半球$t$-设计

中文摘要 AI 辅助

本文在半球$\mathbb{S}^d_+:=\{\mathbf{x}\in\mathbb{R}^{d+1}:\\|\mathbf{x}\\|=1, \mathbf{x}\cdot\mathbf{e}_{d+1}\geq0\}$上引入一组新的$\mathbb{L}_2$-正交多项式，并针对偶数和奇数球面多项式建立了半球上的$\mathbb{L}_1$ Marcinkiewicz-Zygmund不等式。半球$t$-设计提供了半球上等权求积规则，这些规则对不超过$t$次的多项式精确成立。我们通过$\mathbb{L}_2$-正交多项式提出了半球$t$-设计的变分刻画。此外，基于Marcinkiewicz-Zygmund不等式和$\mathbb{L}_2$-正交多项式，我们证明对于每个$n\geq C_d t^d$，半球上存在具有$n$个点的半球$t$-设计，其中$C_d$是仅依赖于$d$的常数。

英文摘要

In this paper, we introduce a new set of $\mathbb{L}_2$-orthogonal polynomials on the hemisphere $\mathbb{S}^d_+:=\{\mathbf{x}\in\mathbb{R}^{d+1}:\|\mathbf{x}\|=1, \mathbf{x}\cdot\mathbf{e}_{d+1}\geq0\}$ and establish $\mathbb{L}_1$ Marcinkiewicz-Zygmund inequalities on the hemisphere for even and odd spherical polynomials. Hemispherical $t$-designs provide equal weight cubature rules on the hemisphere which are exact for polynomials up to degree $t$. We propose variational characterizations of hemispherical $t$-designs by the $\mathbb{L}_2$-orthogonal polynomials. Moreover, based on the Marcinkiewicz-Zygmund inequalities and the $\mathbb{L}_2$-orthogonal polynomials, we prove that for each $n\geq C_d t^d$, there exists a hemispherical $t$-design on the hemisphere with $n$ points, where $C_d$ is a constant depending only on $d$.

发表机构

• The Hong Kong Polytechnic University（香港理工大学）

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