Localization and vector spherical harmonics
In this paper, a localization property for vector spherical harmonics is established. Let \(L^2(S^{d-1}; \mathbb R^d)\) be the space of square-integrable, vector-valued functions with norm \(\|\psi\|^2_{L^2(S^{d-1}; \mathbb R^d)}= \int_{S^{d-1}}|\psi(\omega)|^2dS^{d-1}_\omega\). For any kernel function \(f(s) :(-1, 1)\to\mathbb R\) that satisfies the integrability hypothesis \(f\in L^1(\nu_d)\), where \(d\nu_d(s)=(1-s^2)^{(d-3)/2} \mathbf 1_{(-1,1)}(s)\,ds\) two classes of integral operators local \(L_{f,k(\omega,\sigma)}\) and non-local \(N_{f,k(\omega,\sigma)}\) are associated \[ L_{f,k(\omega,\sigma)}[\psi](\omega)=\int_{S^{d-1}} f(\langle\omega,\sigma\rangle) k(\omega,\sigma)\psi (\omega)dS^{d-1}_\sigma. \] Here the \(d\times d\) matrix kernel \(k(\omega,\sigma)\) stands in for one of the \(d\times d\) matrices \(Id_d\), \(\omega\otimes\omega\), \(\omega\otimes\sigma\), \(\sigma\otimes\omega\), \(\sigma\otimes\sigma\), where \(\otimes\) denotes outer-product and \(Id_d\) denotes the identity matrix. The main result is in Theorem 3.4: Let \(f\in L^1(\nu_d)\) and \(k(\omega, \sigma)\) be defined above, then each of the associated operators \(L_{f,k(\omega,\sigma)}\), \(N_{f,k(\omega,\sigma)}\) generates local basis sequences (Table 1). Moreover, each operator \(L_{f,k(\omega, \sigma)}\), \(N_{f,k(\omega,\sigma)}\) has both \(L_0^2(S^{d-1}; \mathbb R^d)\) and \(L_0^2(S^{d-1}; \mathbb R^d)^\perp\) as invariant subspaces. The proof is based on two lemmas and on the Funk-Hecke formula for scalar spherical harmonics (Theorem 3.1).
scientific article; zbMATH DE number 6506398
scientific article