Needlets liberated

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis Pub Date : 2024-08-02 DOI:10.1016/j.acha.2024.101693

Johann S. Brauchart , Peter J. Grabner , Ian H. Sloan , Robert S. Womersley

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引用次数: 0

Abstract

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 $H^{s}$ , 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.

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释放针头

球面小针由 Narcowich、Petrushev 和 Ward 提出，为球面上的函数提供多项式近似的多分辨率序列。小针构造利用了对给定阶以内多项式精确的积分规则。本文的目的是放宽积分规则的精确性，用 Brauchart、Saff、Sloan 和 Womersley（2014 年）提出的 QMC 设计来代替它们。这种积分规则（此处通过允许非等立方权重进行了概括）提供了与索博廖夫空间精确规则相同的渐近收敛阶数，但更容易在数值上获得。利用这种规则，我们构建了 "广义微分方程"。本文提供了一种误差分析，允许用广义小针取代原始小针，并更广泛地分析了一种混合方案，其中低层次的小针是传统类型的，而新的广义小针用于一些高层次。本文最后还进行了数值实验。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applied and Computational Harmonic Analysis 物理-物理：数学物理

CiteScore

5.40

自引率

4.00%

发文量

审稿时长

22.9 weeks

期刊介绍： Applied and Computational Harmonic Analysis (ACHA) is an interdisciplinary journal that publishes high-quality papers in all areas of mathematical sciences related to the applied and computational aspects of harmonic analysis, with special emphasis on innovative theoretical development, methods, and algorithms, for information processing, manipulation, understanding, and so forth. The objectives of the journal are to chronicle the important publications in the rapidly growing field of data representation and analysis, to stimulate research in relevant interdisciplinary areas, and to provide a common link among mathematical, physical, and life scientists, as well as engineers.