An application of the Hardy-Littlewood Tauberian theorem to harmonic expansion of a complex measure on the sphere
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 \(\mu (\{\zeta\}) = 0\).
scientific article; zbMATH DE number 5904838
scientific article