Some estimates for strong uniform approximation on sphere
Let \(\Sigma_{d-1}\) be the unit sphere in \(R^d\), \(d\geq 3\). The author studies the Ce\`saro means \(\sigma_n^\lambda(f)\) of the critical order \(\lambda={{d-2}\over2}\) for the Fourier-Laplace series of \(f\) from the continuous function space \(C(\Sigma_{d-1})\). Write \[ \begin{aligned} C(F) &=\{f\in C(\Sigma_{d-1}) : E_k(f)\leq F_k, \;k=0, 1, 2,\dots\},\\ H^\omega&=\{f\in C(\Sigma_{d-1}) : \omega(f, t)\leq \omega(t)\}, \end{aligned} \] where \(E_k(f)\) is the best uniform approximation of \(f\) by spherical polynomials of order \(k\), \(\omega(f, t)\) is the modulus of continuity of \(f\), \(\{ F_k\}_{k=0}^\infty\) and \(\omega(t)\) are some related majorants of \(\{ E_k(f)\}_{k=0}^\infty\) and \(\omega(f, t)\), and \(F=\{ F_k\}_{k=0}^\infty\). The following results are proved. If \(\{ F_k\}_{k=0}^\infty\) is a monotone sequence of positive numbers with \(\lim_{k\to\infty} F_k=0\), and if \(q>1\), \(\lambda={{d-2}\over{2}}\), then \[ \sup_{f\in C(F)} \left\| {1\over{n+1}} \sum_{k=0}^n | f-\sigma_k^\lambda(f)| ^q\right\|_C \sim {1\over{n+1}} \sum_{k=0}^n F_k^q \] and if there is a constant \(\beta>0\) such that \[ {1\over{k+1}}\sum_{\nu=0}^k\omega\left( {1\over{\nu+1}}\right)\leq \beta\omega\left({1\over {k+1}}\right), \] then \[ {\sup}_{f\in H^\omega} \left\| {1\over {n+1}}\sum_{k=0}^n | f-\sigma_k^\lambda(f)| ^q\right\|_C \sim {1\over{n+1}}\sum_{k=0}^n \left[\omega\left( {1\over {k+1}}\right)\right]^q. \] It should be observed that these results are well known for the classical Fourier series.
scientific article; zbMATH DE number 2092054