Spherical needlets were introduced by Narcowich, Petrushev, and Ward to provide a multiresolution sequence of polynomial approximations to functions on the sphere. The needlet construction makes use of integration rules that are exact for polynomials up to a given degree. The aim of the present paper is to relax the exactness of the integration rules by replacing them with QMC designs as introduced by Brauchart, Saff, Sloan, and Womersley (2014). Such integration rules (generalised here by allowing non-equal cubature weights) provide the same asymptotic order of convergence as exact rules for Sobolev spaces , but are easier to obtain numerically. With such rules we construct “generalised needlets”. The paper provides an error analysis that allows the replacement of the original needlets by generalised needlets, and more generally, analyses a hybrid scheme in which the needlets for the lower levels are of the traditional kind, whereas the new generalised needlets are used for some number of higher levels. Numerical experiments complete the paper.

中文翻译：

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 释放针头


Narcowich、Petrushev 和 Ward 引入球形针，为球体上的函数提供多项式逼近的多分辨率序列。针结构利用了对于给定次数的多项式来说精确的积分规则。本文的目的是通过用 Brauchart、Saff、Sloan 和 Womersley (2014) 引入的 QMC 设计替换积分规则来放宽积分规则的精确性。这种积分规则（此处通过允许不相等的体积权重进行概括）提供了与 Sobolev 空间的精确规则相同的渐近收敛顺序，但更容易通过数值获得。根据这样的规则，我们构建了“通用针”。本文提供了一种误差分析，允许用广义针替换原始针，更一般地说，分析了一种混合方案，其中较低级别的针是传统类型的，而新的广义针用于某些数量的针。更高级别的。数值实验完成了论文。