Suppose G is a n-dimensional compact connected semisimple Lie group and D_R is the spherical Dirichlet kernel on G. We prove the existence of a positive constant K such that ||D_R||_1 ≥ k R^{(n-1)/2}. This complements the known result .|D_R||_1 ≤ k R^{(n-1)/2}. We also prove that for a polyhedral Dirichlet kernel D_N the above inequalities hold with N^p in place of R^{(n-1)/2} (p is the number of positive roots of G).
Sharp estimates for Lebesgue constants on compact Lie group
GIULINI, SAVERIO;
1986-01-01
Abstract
Suppose G is a n-dimensional compact connected semisimple Lie group and D_R is the spherical Dirichlet kernel on G. We prove the existence of a positive constant K such that ||D_R||_1 ≥ k R^{(n-1)/2}. This complements the known result .|D_R||_1 ≤ k R^{(n-1)/2}. We also prove that for a polyhedral Dirichlet kernel D_N the above inequalities hold with N^p in place of R^{(n-1)/2} (p is the number of positive roots of G).
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