Discretized Newman-Shapiro operators and Jackson's inequality on the sphere
The author takes a quadrature formula on the \(r\)-dimensional sphere, \(r>2\), that has positive weights and which is exact for spherical polynomials of degree \(2\nu+1\), and applies it to what the author refers to as the Newman-Shapiro operator on that sphere. Denoting the resulting operator \(L_\mu\), where \(\mu=2\nu\), he then proves that there exists an absolute constant \(k\) depending only on the dimension \(r\), such that for every function \(F\in C(S^{r-1})\), we have \(\|F-L_\mu F\|\leq k\omega_F(1/\mu)\), where \(\|\cdot\|\) denotes the uniform norm on the sphere and \(\omega_F(t):=\sup\{|F(x)-F(y)|:x,y\in S^{r-1},x\cdot y\geq\cos t\}\). It should be stressed that \(k\) is independent of the quadrature formula provided it possesses the above requirements. The proof involves properties of the Gegenbauer polynomials.
scientific article; zbMATH DE number 1383728
scientific article