Sharp estimates for Lebesgue constants on compact Lie groups (Q1115540)

Suppose G is an n-dimensional compact connected semisimple Lie group and \(D_ R\) is the spherical Dirichlet kernel on G. A lower bound for the \(L^ 1\)-norm of \(D_ R\) is given in the way \(\| D_ R\|_ 1\geq K R^{(n-1)/2}\), where K is a constant non-depending on R. This complements the known result \(\| D_ R\|_ 1\leq H R^{(n-1)/2}\). It is also shown that for a polyhedral Dirichlet kernel \(D_ N\) the above inequalities hold with \(N^ p\) in place of \(R^{(n-1)/2}\), where p is the number of positive roots of G.