Splitting methods for SU(N) loop approximation Original Research Article

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 View the MathML sourceSU(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 View the MathML sourceSU(N)-loops, N≥2N≥2. In particular, using representations via the exponential map and first order splitting methods, we prove that the best approximation of an View the MathML sourceSU(N)-loop belonging to a Hölder–Zygmund class View the MathML sourceLipα, α>1/2α>1/2, by a polynomial View the MathML sourceSU(N)-loop of degree ≤n≤n is of the order O(n−α/(1+α))(n−α/(1+α)) as n→∞n→∞. Although this approximation rate is not considered final, to our knowledge it is the first general, nontrivial result of this type.