Close-to-optimal bounds for image loop approximation

In Oswald and Shingel (2009) [6], we proved an asymptotic View the MathML sourceO(n−α/(α+1)) bound for the approximation of View the MathML sourceSU(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 View the MathML sourceSU(N) loops into products of constant View the MathML sourceSU(N) matrices and loops of the form View the MathML sourceeA(t) where A(t)A(t) are essentially View the MathML sourcesu(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 View the MathML sourceO(n−(α−ϵ)) bounds for α>1α>1, where ϵ>0ϵ>0 can be chosen arbitrarily small.