Orthogonal Developments on Compact Symmetric Homogeneous Manifolds of Rank 1

Sharp asymptotic for the norms of Fourier projections on compact homogeneous manifolds of rank 1 are established. These results extend sharp asymptotic estimates found by Fejer in the case of the circle in 1910 and then by Gronwall in 1914 in the case of the 2-dimensional sphere. As an application of these results we give sharp asymptotic for the rate of convergence of Fourier sums on a wide range of sets of multipliers.