  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>1. 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 Hoelder-Zygmund class Lip_alpha with alpha>1/2 by a polynomial
SU(N)-loop of degree n is of the order O(n^{-\alpha/(1+\alpha)}) as n tends to
infinity. 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.
