Sharp estimates for Lebesgue constants on compact Lie groups

AbstractSuppose G is a n-dimensional compact connected semisimple Lie group and DR is the spherical Dirichlet kernel on G. We prove the existence of a positive constant K such that ∥DR∥1 ⩾ KR(n − 1)2 This complements the known result ∥DR∥1 ⩽ HR(n − 1)2. We also prove that for a polyhedral Dirichlet kernel DN the above inequalities hold with Np in place of R(n − 1)2 (p is the number of positive roots of G)