Averages on caps of \(S^{d-1}\)
Let \(S^{d-1}\) be the unit sphere of the Euclidean space \(\mathbb R^d\), \(d\geq 3.\) For given \( y\in S^{d-1}\) and \( t\in (0,\pi)\) let \(C(y,t):=\{ x\in S^{d-1}:\cos t\leq\langle x,y\rangle\leq 1\}\) be a cap of \(S^{d-1}\) centered at \(y\) of angle \(\leq t.\) For Lebesgue integrable function \(f\) on \(S^{d-1}\) the average of \(f\) on \(C(y,t)\) is given by \[ B_t(f,y):=\frac 1{\Phi(t)}\int_{C(y,t)}f(x) d\sigma(x), \] where \(d\sigma\) is the measure on \(S^{d-1}\) and \(\Phi\) is a normalizing factor. The authors show that for the \(p\)-integrable function \(f\) on \(S^{d-1}\), \(1\leq p\leq\infty\) the rate of convergence \(\|f-B_tf\|_{L_p(S^{d-1})}\) is equivalent to some \(K\)-functional which is constructed in terms of Laplace-Beltrami operator on \(S^{d-1}.\) Analogous results are proved for averages on the rim of the caps and for so called Steklov-type means.
scientific article; zbMATH DE number 1506533