Splitting methods for SU(N) loop approximation

The problem of finding the correct asymptotic rate of approximation by polynomial loops in dependence of the smoothness of the elements of a loop group seems not well-understood in general. For matrix Lie groups such as SU(N), it can be viewed as a problem of nonlinearly constrained trigonometric approximation. Motivated by applications to optical FIR filter design and control, we present some initial results for the case of SU(N)-loops, N ≥ 2. In particular, using representations via the exponential map and ideas from splitting methods, we prove that the best approximation of an SU(N)-loop belonging to a Hölder-Zygmund class Lipα, α> 1/2, by a polynomial SU(N)-loop of degree ≤ n is of the order O(n−α/(1+α)) as n → ∞. Although this approximation rate is not considered final (and can be improved in special cases), to our knowledge it is the first general, nontrivial result of this type.