An application of the Hardy-Littlewood Tauberian theorem to harmonic expansion of a complex measure on the sphere
type: Item
instance of: scholarly article
title: An application of the Hardy-Littlewood Tauberian theorem to harmonic expansion of a complex measure on the sphere
zbMATH Open document ID: 1230.42036
author: Yifei Pan
author: Mei Wang
published in: Real Analysis Exchange
publication date: 2011-06-07T00:00:00Z
review text: Let \(Z_m (\zeta,\cdot)\) denote the zonal harmonic of order \(m\) with pole \(\zeta\). Each complex measure \(\mu\) on the unit sphere \(S^{n-1}\) in \(\mathbb{R}^n\) has a spherical harmonic expansion \(\sum_{m=0}^\infty p_m\) with \[ p_m (\zeta) = \int_{S^{n-1}} Z_m (\zeta,\eta) d\mu (\eta) \qquad (\zeta \in S^{n-1}) \] being a spherical harmonic of degree \(m\). The main theorem, which is essentially an application of the Hardy-Littlewood Tauberian Theorem, states that \[ \sum_{m=0}^N p_m (\zeta) \sim \frac{2}{(n-1)!}\, \mu (\{\zeta\}) N^{n-1} \qquad (N\to\infty) \] if \(\mu (\{\zeta\}) > 0\) and \[ \sum_{m=0}^N p_m (\zeta) = o(N^{n-1}) \qquad (N\to\infty) \] if \(\mu (\{\zeta\}) = 0\).
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 33C55
zbMATH DE Number: 5904838
zbMATH Keywords: Tauberian theorems
zbMATH Keywords: spherical harmonics
zbMATH Keywords: zonal functions
reviewed by: Jürgen Müller
MaRDI profile type: Publication
OpenAlex ID: W1500611437
DOI: 10.14321/REALANALEXCH.35.2.0517
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description: scientific article; zbMATH DE number 5904838