Close-to-optimal bounds for SU(N) loop approximation

In Oswald and Shingel (2009) [6], we proved an asymptotic O(n−α/(α+1)) bound for the approximation of SU(N) loops (N≥2N≥2) with Lipschitz smoothness α>1/2α>1/2 by polynomial loops of degree ≤n≤n. The proof combined factorizations of SU(N) loops into products of constant SU(N) matrices and loops of the form eA(t) where A(t)A(t) are essentially su(2) loops preserving the Lipschitz smoothness, and the careful estimation of errors induced by approximating matrix exponentials by first-order splitting methods. In the present note we show that using higher order splitting methods allows us to improve the above suboptimal result to close-to-optimal O(n−(α−ϵ)) bounds for α>1α>1, where ϵ>0ϵ>0 can be chosen arbitrarily small.