Jackson-type inequalities on high-dimensional spheres (Q6630744)

Let \(S^d\) denote the unit sphere of \(R^{d+1}\) equipped with the surface Lebesgue measure normalized by \(\int_{S^d}1 d\sigma(x)=1\). Given \(1\leq p < \infty\), denote by \(L^p(S^d)\) the Lebesgue \(L^p\) space equipped with the norm \(\| f\|_p:=\left ( \int_{S^d}| f (x)|^p d\sigma(x)\right)^{1/p}\). When \(p=\infty\) let \(L^{\infty}(S^d)\) denote the space \(C(S^d)\) of all continuous functions on \(S^d\) with the uniform norm.\N\NA spherical polynomial of degree at most \(n\) on \(S^d\) is the restriction on \(S^d\) of an algebraic polynomial in \(d + 1\) variables of total degree at most \(n\). The main result of the paper is the following \N\NTheorem. For \(1\leq p \leq \infty\) and \(r\in N\), there exists a constant \(C_r>0\) depending only on \(r\) such that for all \(f\in L^p(S^d)\) and \(n=1, 2, \dots\), \[ E_n(f)_{p} \leq C_r\omega^r \left( f, \frac{(d+1)^3}{n} \right)_{p}, \] and \[ \omega^r \left( f, n^{-1} \right)_{p}\leq C_r n^{-r} \sum_{k=1}^{n}k^{r-1}E_{k-1}(f)_p, \] where \(E_n(f)_p\) is the best approximation of \(f\) by the space of spherical polynomials of total degree at most n in the \(L^p(S^d)\) norm and \(\omega ^r \left( f, \cdot \right)_{p}\) is the \(r\)-th order modulus of smoothness of \(f\in L^p(S^d)\) in the space \(L^p(S^d)\). \N\NWe note that the first inequality in the Theorem above in the case of \(p=\infty\) and \(r=1\) was obtained in \N[in: Über Approximationstheorie. Proceedings of the conference held in the Mathematical Research Institute at Oberwolfach, Black Forest, August 4--10, 1963. Basel: Springer. 208--219 (1964; Zbl 0158.05506)] by \textit{D. J. Newman} and \textit{H. S. Shapiro}.\N\NSimilar results on the equivalence of the \(K\)-functional and modulus of smoothness are also established.