Lebesgue constants on compact manifolds

Sharp asymptotic for norms of Fourier projections on two-point homogeneous manifolds (the real sphere, the real, complex and quaternionic projective spaces and the Cayley elliptic plain) are established. These results extend sharp asymptotic estimates found by Fejer on the circle (in 1910) and then by Gronwall on the two-dimensional sphere (in 1914). As an application of these results we give solution of the problem of Kolmogorov on sharp asymptotic for the rate of convergence of Fourier sums on a wide range of sets of multiplier operators.