On the Lebesgue constants

Abstract

UDC 517.5
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 spheres $\mathbb{S}^{d},$ complex $\mathrm{P}^{d}(\mathbb{C})$ and quaternionic $\mathrm{P}^{d}(\mathbb{H})$ projective spaces, and the Cayley elliptic plane $\mathrm{P}^{16}(\mathrm{Cay}).$ In particular, these results extend sharp asymptotic found by Fejer in the case of $\mathbb{S}^{1}$ in 1910 and by Gronwall in 1914 in the case of $\mathbb{S}^{2}$ .