Orthogonal Developments on Compact Symmetric Homogeneous Manifolds of Rank 1

Sharp asymptotics of the norms of Fourier projections on compact homogeneous manifolds $M^d$ of rank 1, i.e., on $S^d$, $P^d(R)$, $P^d(C)$, $P^d(H)$, $P^{16}(Cay)$ are established. These results extend sharp asymptotic estimates found by Fejer [ 5] in the case of $S$ in 1910 and then by Gronwall [ 7] in 1914 in the case of $S^2$.