On approximation by spherical zonal translation networks based on Bochner-Riesz means (Q2577602)

Based on the Bochner-Riesz means of Fourier-Laplace series on the unit sphere \(S^q\) of the Euclidian space of dimension \(q+1, (q\geq 2)\), the authors constructed a kind of zonal translation operators \(M_{R,\varphi}\) where the action function \(\varphi\in L^p_{w_q} (1\leq p\leq \infty)\) with \(w_q\) being the Jacobi weight \(w_{{q\over 2},{q\over 2}}\). Under certain conditions it was proved that \[ \lim_{R\to\infty}\| f- M_{R,\varphi}(f)\| _{L^p(S^q)}=0\;\forall f\in L^p(S^q). \] The reference [\textit{H. N. Mhaskar, F. J. Narcowich} and \textit{J. D. Ward}, Adv. Comput. Math. 11, No. 2--3, 121--137 (1999; Zbl 0939.41012)] provided prerequisite knowledge on the spherical translation networks.