Close-to-optimal bounds for [formula omitted] loop approximation

Abstract

In Oswald and Shingel (2009) [6], we proved an asymptotic O ( n − α / ( α + 1 ) ) bound for the approximation of SU ( N ) loops ( N ≥ 2 ) with Lipschitz smoothness α > 1 / 2 by polynomial loops of degree ≤ n . The proof combined factorizations of SU ( N ) loops into products of constant SU ( N ) matrices and loops of the form e A ( t ) where 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 > 1 , where ϵ > 0 can be chosen arbitrarily small.