The Lebesgue constants on projective spaces

We give the solution of a classical problem of Approximation Theory on sharp asymptotic of the Lebesgue constants or norms of the Fourier-Laplace projections on the real projective spaces P-d (R). In particular, these results extend sharp asymptotic found by Fejer [2] in the case of S-1 in 1910 and by Gronwall [4] in 1914 in the case of S-2. The case of spheres, S-d, complex and quaternionic projective spaces, P-d(C), P-d(H) and the Cayley elliptic plane P-16 (Cay) was considered by Kushpel [8].