On approximation of Bochner-Riesz means on the unit sphere (Q1200066)

For any function \(f\) on the unit sphere \(\Sigma_ n\) in \((n+1)\)- dimensional Euclidean space consider the expansion in spherical harmonics \(f\sim \sum Y_ k\) and the Bochner-Riesz (B-R) means of index \(\delta\) \[ S_ R^ \delta=\sum_{k<R}(1-k^ 2/R^ 2)^ \delta Y_ k. \] The index \(\delta=(n-1)/2\) is critical for B-R summability. The paper contains the estimate \[ \| S_ R^ \delta-f\|=O(\log R)\omega(f,R^{-1})\quad\text{as}\quad R\to\infty \] for functions \(f\in C(\Sigma_ n)\) in the uniform norm for the critical index \(\delta\). The function \(\omega\) is a sort of modulus of continuity. This estimate can be improved if \(f\) satisfies additional smoothness hypotheses. The paper contains also analogous estimates in the \(L^ 2\)-norm for Riesz potentials \(f\) and any positive index \(\delta\). There are no proofs given.