Jackson's theorem for compact connected Lie groups

Abstract

We prove that for f ϵ E = C( G) or L p ( G), 1 ⩽ p < ∞, where G is any compact connected Lie group, and for n ⩾ 1, there is a trigonometric polynomial t n on G of degree ⩽ n so that ‖ f − t n ‖ E ⩽ C r ω r ( n −1, f). Here ω r ( t, f) denotes the rth modulus of continuity of f. Using this and sharp estimates of the Lebesgue constants recently obtained by Giulini and Travaglini, we obtain “best possible” criteria for the norm convergence of the Fourier series of f.