Unbounded Lebesgue constants on compact groups

Let G be a compact group with dual object Γ. A decomposition of Γ into pairwise disjoint finite sets provides a way of defining partial sums of Fourier series on G. We employ a theorem of Olevskii to present a condition on such a decomposition which, if satisfied, guarantees that the corresponding Lebesgue constants are unbounded. We apply this to certain summation methods on tori and compact connected semisimple Lie groups.